TPTP Problem File: SLH0904^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Interpolation_Polynomials_HOL_Algebra/0001_Lagrange_Interpolation/prob_00260_010101__17373200_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1556 ( 256 unt; 279 typ; 0 def)
% Number of atoms : 4551 (1299 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 22634 ( 288 ~; 44 |; 141 &;19302 @)
% ( 0 <=>;2859 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 9 avg)
% Number of types : 27 ( 26 usr)
% Number of type conns : 677 ( 677 >; 0 *; 0 +; 0 <<)
% Number of symbols : 254 ( 253 usr; 12 con; 0-4 aty)
% Number of variables : 3262 ( 105 ^;3078 !; 79 ?;3262 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:38:24.799
%------------------------------------------------------------------------------
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member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_member_001tf__a,type,
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thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_S,type,
s: set_a ).
thf(sy_v_p____,type,
p: list_a ).
thf(sy_v_s____,type,
s2: a ).
% Relevant facts (1276)
thf(fact_0_d,axiom,
member_a @ s2 @ s ).
% d
thf(fact_1_factorial__domain__axioms,axiom,
ring_f5272581269873410839in_a_b @ r ).
% factorial_domain_axioms
thf(fact_2_local_Ofield__axioms,axiom,
field_a_b @ r ).
% local.field_axioms
thf(fact_3_assms_I1_J,axiom,
finite_finite_a @ s ).
% assms(1)
thf(fact_4_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_5_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_6_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_7_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_8_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_9_calculation,axiom,
( ( eval_a_b @ r @ p @ s2 )
= ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ s2 ) @ s ) ) ).
% calculation
thf(fact_10_finprod__one__eqI,axiom,
! [A: set_list_a,F: list_a > a] :
( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( ( F @ X )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_11_finprod__one__eqI,axiom,
! [A: set_list_list_a,F: list_list_a > a] :
( ! [X: list_list_a] :
( ( member_list_list_a @ X @ A )
=> ( ( F @ X )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro5500967685102550467list_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_12_finprod__one__eqI,axiom,
! [A: set_a,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( F @ X )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_13_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_14_subring__props_I2_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K ) ) ).
% subring_props(2)
thf(fact_15_zero__divides,axiom,
! [A2: a] :
( ( factor8216151070175719842xt_a_b @ r @ ( zero_a_b @ r ) @ A2 )
= ( A2
= ( zero_a_b @ r ) ) ) ).
% zero_divides
thf(fact_16_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_17_add_Ofinprod__one__eqI,axiom,
! [A: set_a,F: a > a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( F @ X )
= ( zero_a_b @ r ) ) )
=> ( ( finsum_a_b_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_18_add_Ofinprod__one__eqI,axiom,
! [A: set_list_a,F: list_a > a] :
( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( ( F @ X )
= ( zero_a_b @ r ) ) )
=> ( ( finsum_a_b_list_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_19_add_Ofinprod__one__eqI,axiom,
! [A: set_list_list_a,F: list_list_a > a] :
( ! [X: list_list_a] :
( ( member_list_list_a @ X @ A )
=> ( ( F @ X )
= ( zero_a_b @ r ) ) )
=> ( ( finsum1795837918752241516list_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% add.finprod_one_eqI
thf(fact_20_p__def,axiom,
( p
= ( lagran9092808442999052491ux_a_b @ r @ s ) ) ).
% p_def
thf(fact_21_subring__props_I3_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K ) ) ).
% subring_props(3)
thf(fact_22_finsum__infinite,axiom,
! [A: set_a,F: a > a] :
( ~ ( finite_finite_a @ A )
=> ( ( finsum_a_b_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% finsum_infinite
thf(fact_23_finsum__infinite,axiom,
! [A: set_list_a,F: list_a > a] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finsum_a_b_list_a @ r @ F @ A )
= ( zero_a_b @ r ) ) ) ).
% finsum_infinite
thf(fact_24_finprod__infinite,axiom,
! [A: set_list_a,F: list_a > a] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finpro6052973074229812797list_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_25_finprod__infinite,axiom,
! [A: set_a,F: a > a] :
( ~ ( finite_finite_a @ A )
=> ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_26_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
lagran9092808442999052491ux_a_b = lagran9092808442999052491ux_a_b ).
% ring.lagrange_basis_polynomial_aux.cong
thf(fact_27_ring__primeI,axiom,
! [P: a] :
( ( P
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P )
=> ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% ring_primeI
thf(fact_28_abelian__monoid_Ofinsum__infinite,axiom,
! [G: partia2175431115845679010xt_a_b,A: set_a,F: a > a] :
( ( abelian_monoid_a_b @ G )
=> ( ~ ( finite_finite_a @ A )
=> ( ( finsum_a_b_a @ G @ F @ A )
= ( zero_a_b @ G ) ) ) ) ).
% abelian_monoid.finsum_infinite
thf(fact_29_abelian__monoid_Ofinsum__infinite,axiom,
! [G: partia2175431115845679010xt_a_b,A: set_list_a,F: list_a > a] :
( ( abelian_monoid_a_b @ G )
=> ( ~ ( finite_finite_list_a @ A )
=> ( ( finsum_a_b_list_a @ G @ F @ A )
= ( zero_a_b @ G ) ) ) ) ).
% abelian_monoid.finsum_infinite
thf(fact_30_abelian__monoid_Ofinsum__infinite,axiom,
! [G: partia2670972154091845814t_unit,A: set_a,F: a > list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ~ ( finite_finite_a @ A )
=> ( ( finsum7322697649718157656unit_a @ G @ F @ A )
= ( zero_l4142658623432671053t_unit @ G ) ) ) ) ).
% abelian_monoid.finsum_infinite
thf(fact_31_abelian__monoid_Ofinsum__infinite,axiom,
! [G: partia2670972154091845814t_unit,A: set_list_a,F: list_a > list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ~ ( finite_finite_list_a @ A )
=> ( ( finsum8721804980556663006list_a @ G @ F @ A )
= ( zero_l4142658623432671053t_unit @ G ) ) ) ) ).
% abelian_monoid.finsum_infinite
thf(fact_32_finprod__zero__iff,axiom,
! [A: set_list_list_a,F: list_list_a > a] :
( ( finite1660835950917165235list_a @ A )
=> ( ! [A3: list_list_a] :
( ( member_list_list_a @ A3 @ A )
=> ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro5500967685102550467list_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_33_finprod__zero__iff,axiom,
! [A: set_list_a,F: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ A )
=> ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro6052973074229812797list_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_34_finprod__zero__iff,axiom,
! [A: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_35_one__divides,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ A2 ) ) ).
% one_divides
thf(fact_36_zero__is__irreducible__iff__field,axiom,
( ( irredu6211895646901577903xt_a_b @ r @ ( zero_a_b @ r ) )
= ( field_a_b @ r ) ) ).
% zero_is_irreducible_iff_field
thf(fact_37_divides__zero,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A2 @ ( zero_a_b @ r ) ) ) ).
% divides_zero
thf(fact_38_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_39_Ring_Oone__not__zero,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_40_Ring_Oone__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_41_Ring_Oone__not__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_42_subfieldE_I6_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% subfieldE(6)
thf(fact_43_subfieldE_I6_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% subfieldE(6)
thf(fact_44_const__term__def,axiom,
! [P: list_a] :
( ( const_term_a_b @ r @ P )
= ( eval_a_b @ r @ P @ ( zero_a_b @ r ) ) ) ).
% const_term_def
thf(fact_45_noetherian__domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_n4045954140777738665in_a_b @ R )
=> ( ring_n3639167112692572309ng_a_b @ R ) ) ).
% noetherian_domain.axioms(1)
thf(fact_46_telescopic__base__dim_I1_J,axiom,
! [K: set_a,F2: set_a,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( subfield_a_b @ F2 @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ F2 )
=> ( ( embedd8708762675212832759on_a_b @ r @ F2 @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K @ E ) ) ) ) ) ).
% telescopic_base_dim(1)
thf(fact_47_carrier__is__subfield,axiom,
subfield_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subfield
thf(fact_48_divides__trans,axiom,
! [A2: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A2 @ B )
=> ( ( factor8216151070175719842xt_a_b @ r @ B @ C )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A2 @ C ) ) ) ) ).
% divides_trans
thf(fact_49_const__term__not__zero,axiom,
! [P: list_a] :
( ( ( const_term_a_b @ r @ P )
!= ( zero_a_b @ r ) )
=> ( P != nil_a ) ) ).
% const_term_not_zero
thf(fact_50_ring__primeE_I1_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( P
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_51_ring__primeE_I3_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( prime_a_ring_ext_a_b @ r @ P ) ) ) ).
% ring_primeE(3)
thf(fact_52_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_53_assms_I2_J,axiom,
ord_less_eq_set_a @ s @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(2)
thf(fact_54_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_55_divides__refl,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A2 @ A2 ) ) ).
% divides_refl
thf(fact_56_minus__closed,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_57_r__right__minus__eq,axiom,
! [A2: a,B: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A2 @ B )
= ( zero_a_b @ r ) )
= ( A2 = B ) ) ) ) ).
% r_right_minus_eq
thf(fact_58_field_Ocarrier__is__subfield,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( subfie4339374749748326226t_unit @ ( partia141011252114345353t_unit @ R ) @ R ) ) ).
% field.carrier_is_subfield
thf(fact_59_field_Ocarrier__is__subfield,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( subfield_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% field.carrier_is_subfield
thf(fact_60_field_Ocarrier__is__subfield,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( subfie1779122896746047282t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).
% field.carrier_is_subfield
thf(fact_61_field_Ocarrier__is__subfield,axiom,
! [R: partia2956882679547061052t_unit] :
( ( field_1861437471013600865t_unit @ R )
=> ( subfie4546268998243038636t_unit @ ( partia2464479390973590831t_unit @ R ) @ R ) ) ).
% field.carrier_is_subfield
thf(fact_62_mem__Collect__eq,axiom,
! [A2: a,P2: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A2: list_a,P2: list_a > $o] :
( ( member_list_a @ A2 @ ( collect_list_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A2: list_list_a,P2: list_list_a > $o] :
( ( member_list_list_a @ A2 @ ( collect_list_list_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_65_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X2: list_a] : ( member_list_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X2: list_list_a] : ( member_list_list_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_68_abelian__monoid_Ozero__closed,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ G )
=> ( member_a @ ( zero_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_69_abelian__monoid_Ozero__closed,axiom,
! [G: partia2670972154091845814t_unit] :
( ( abelia226231641709521465t_unit @ G )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_70_abelian__monoid_Ozero__closed,axiom,
! [G: partia2956882679547061052t_unit] :
( ( abelia3641329199688042803t_unit @ G )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ G ) @ ( partia2464479390973590831t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_71_abelian__monoidE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_72_abelian__monoidE_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( abelia226231641709521465t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_73_abelian__monoidE_I2_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( abelia3641329199688042803t_unit @ R )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_74_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_75_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_76_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_77_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_78_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_79_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( member_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_80_ring__prime__def,axiom,
( ring_ring_prime_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,A4: a] :
( ( A4
!= ( zero_a_b @ R2 ) )
& ( prime_a_ring_ext_a_b @ R2 @ A4 ) ) ) ) ).
% ring_prime_def
thf(fact_81_ring__prime__def,axiom,
( ring_r6430282645014804837t_unit
= ( ^ [R2: partia2670972154091845814t_unit,A4: list_a] :
( ( A4
!= ( zero_l4142658623432671053t_unit @ R2 ) )
& ( prime_2011924034616061926t_unit @ R2 @ A4 ) ) ) ) ).
% ring_prime_def
thf(fact_82_semiring_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( abelian_monoid_a_b @ R ) ) ).
% semiring.axioms(1)
thf(fact_83_semiring_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( abelia226231641709521465t_unit @ R ) ) ).
% semiring.axioms(1)
thf(fact_84_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_85_isgcd__divides__l,axiom,
! [A2: a,B: a] :
( ( factor8216151070175719842xt_a_b @ r @ A2 @ B )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ A2 @ A2 @ B ) ) ) ) ).
% isgcd_divides_l
thf(fact_86_isgcd__divides__r,axiom,
! [B: a,A2: a] :
( ( factor8216151070175719842xt_a_b @ r @ B @ A2 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ B @ A2 @ B ) ) ) ) ).
% isgcd_divides_r
thf(fact_87_primeness__condition,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P )
= ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% primeness_condition
thf(fact_88_ring__irreducibleE_I2_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( irredu6211895646901577903xt_a_b @ r @ R3 ) ) ) ).
% ring_irreducibleE(2)
thf(fact_89_ring__irreducibleE_I1_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( R3
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_90_is__root__def,axiom,
! [P: list_a,X3: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P @ X3 )
= ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P @ X3 )
= ( zero_a_b @ r ) )
& ( P != nil_a ) ) ) ).
% is_root_def
thf(fact_91_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A3
!= ( zero_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A3 @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_92_finite__dimension__imp__subalgebra,axiom,
! [K: set_a,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ( embedd9027525575939734154ra_a_b @ K @ E @ r ) ) ) ).
% finite_dimension_imp_subalgebra
thf(fact_93_s_Oring_Ozero__closed,axiom,
member_a @ ( eval_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ s2 ) @ ( partia707051561876973205xt_a_b @ r ) ).
% s.ring.zero_closed
thf(fact_94_x_Osemiring__axioms,axiom,
semiri2871908745932252451t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.semiring_axioms
thf(fact_95_x_Oabelian__monoid__axioms,axiom,
abelia226231641709521465t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.abelian_monoid_axioms
thf(fact_96_x_Osubring__props_I3_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) ) ).
% x.subring_props(3)
thf(fact_97_m__lcomm,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X3 @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_98_m__comm,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X3 ) ) ) ) ).
% m_comm
thf(fact_99_m__assoc,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X3 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_100_subring__props_I6_J,axiom,
! [K: set_a,H1: a,H2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H2 @ K )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H2 ) @ K ) ) ) ) ).
% subring_props(6)
thf(fact_101_x_Oring__primeI,axiom,
! [P: list_a] :
( ( P
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.ring_primeI
thf(fact_102_x_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_a,F: a > list_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ( F @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.finprod_one_eqI
thf(fact_103_x_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_list_a,F: list_a > list_a] :
( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( ( F @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.finprod_one_eqI
thf(fact_104_x_Oadd_Ofinprod__one__eqI,axiom,
! [A: set_list_list_a,F: list_list_a > list_a] :
( ! [X: list_list_a] :
( ( member_list_list_a @ X @ A )
=> ( ( F @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( finsum666616272051086308list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.finprod_one_eqI
thf(fact_105_x_Osubring__props_I2_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) ) ).
% x.subring_props(2)
thf(fact_106_eval__in__carrier__2,axiom,
! [X3: list_a,Y: a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier_2
thf(fact_107_m__rcancel,axiom,
! [A2: a,B: a,C: a] :
( ( A2
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B @ A2 )
= ( mult_a_ring_ext_a_b @ r @ C @ A2 ) )
= ( B = C ) ) ) ) ) ) ).
% m_rcancel
thf(fact_108_m__lcancel,axiom,
! [A2: a,B: a,C: a] :
( ( A2
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A2 @ B )
= ( mult_a_ring_ext_a_b @ r @ A2 @ C ) )
= ( B = C ) ) ) ) ) ) ).
% m_lcancel
thf(fact_109_integral__iff,axiom,
! [A2: a,B: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A2 @ B )
= ( zero_a_b @ r ) )
= ( ( A2
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_110_local_Ointegral,axiom,
! [A2: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A2 @ B )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A2
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_111_subring__props_I1_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_112_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X )
= X ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_113_inv__unique,axiom,
! [Y: a,X3: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y @ X3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% inv_unique
thf(fact_114_pprimeE_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( P != nil_a ) ) ) ) ).
% pprimeE(1)
thf(fact_115_divides__prod__r,axiom,
! [A2: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A2 @ B )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A2 @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ).
% divides_prod_r
thf(fact_116_divides__prod__l,axiom,
! [A2: a,B: a,C: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A2 @ B )
=> ( factor8216151070175719842xt_a_b @ r @ A2 @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ) ).
% divides_prod_l
thf(fact_117_local_Odivides__mult,axiom,
! [A2: a,C: a,B: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A2 @ B )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A2 ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% local.divides_mult
thf(fact_118_subalgebra__in__carrier,axiom,
! [K: set_a,V: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_119_carrier__is__subalgebra,axiom,
! [K: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_120_b,axiom,
member_list_a @ p @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% b
thf(fact_121_subalbegra__incl__imp__finite__dimension,axiom,
! [K: set_a,E: set_a,V: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ( ord_less_eq_set_a @ V @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K @ V ) ) ) ) ) ).
% subalbegra_incl_imp_finite_dimension
thf(fact_122_lagrange__aux__poly,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( lagran9092808442999052491ux_a_b @ r @ S ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% lagrange_aux_poly
thf(fact_123_x_Oone__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.one_closed
thf(fact_124_m__closed,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_125_x_Ozero__closed,axiom,
member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.zero_closed
thf(fact_126_r__null,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_127_l__null,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X3 )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_128_r__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( one_a_ring_ext_a_b @ r ) )
= X3 ) ) ).
% r_one
thf(fact_129_l__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X3 )
= X3 ) ) ).
% l_one
thf(fact_130_divides__mult__rI,axiom,
! [A2: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A2 @ B )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A2 @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% divides_mult_rI
thf(fact_131_divides__mult__lI,axiom,
! [A2: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A2 @ B )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A2 ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% divides_mult_lI
thf(fact_132_x_Ofinsum__infinite,axiom,
! [A: set_a,F: a > list_a] :
( ~ ( finite_finite_a @ A )
=> ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.finsum_infinite
thf(fact_133_x_Ofinsum__infinite,axiom,
! [A: set_list_a,F: list_a > list_a] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.finsum_infinite
thf(fact_134_s_Oring_Ohom__closed,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_a @ ( eval_a_b @ r @ X3 @ s2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% s.ring.hom_closed
thf(fact_135_s_Oring_Ohom__one,axiom,
( ( eval_a_b @ r @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ s2 )
= ( one_a_ring_ext_a_b @ r ) ) ).
% s.ring.hom_one
thf(fact_136_s_Oring_Ohom__zero,axiom,
( ( eval_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ s2 )
= ( zero_a_b @ r ) ) ).
% s.ring.hom_zero
thf(fact_137_subfieldE_I4_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b,K1: a,K2: a] :
( ( subfield_a_b @ K @ R )
=> ( ( member_a @ K1 @ K )
=> ( ( member_a @ K2 @ K )
=> ( ( mult_a_ring_ext_a_b @ R @ K1 @ K2 )
= ( mult_a_ring_ext_a_b @ R @ K2 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_138_subfieldE_I4_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit,K1: list_a,K2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_a @ K1 @ K )
=> ( ( member_list_a @ K2 @ K )
=> ( ( mult_l7073676228092353617t_unit @ R @ K1 @ K2 )
= ( mult_l7073676228092353617t_unit @ R @ K2 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_139_subfieldE_I5_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b,K1: a,K2: a] :
( ( subfield_a_b @ K @ R )
=> ( ( member_a @ K1 @ K )
=> ( ( member_a @ K2 @ K )
=> ( ( ( mult_a_ring_ext_a_b @ R @ K1 @ K2 )
= ( zero_a_b @ R ) )
=> ( ( K1
= ( zero_a_b @ R ) )
| ( K2
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_140_subfieldE_I5_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit,K1: list_a,K2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_a @ K1 @ K )
=> ( ( member_list_a @ K2 @ K )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ K1 @ K2 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( K1
= ( zero_l4142658623432671053t_unit @ R ) )
| ( K2
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_141_subfieldE_I3_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subfieldE(3)
thf(fact_142_subfieldE_I3_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subfieldE(3)
thf(fact_143_subfieldE_I3_J,axiom,
! [K: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subfie4546268998243038636t_unit @ K @ R )
=> ( ord_le8488217952732425610list_a @ K @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% subfieldE(3)
thf(fact_144_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_145_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ X3 @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_146_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ R @ X3 @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_147_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R @ X3 @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_148_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X3 @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ R @ X3 @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_149_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X3 @ Y ) @ Z )
= ( mult_l4853965630390486993t_unit @ R @ X3 @ ( mult_l4853965630390486993t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_150_Ring_Ointegral,axiom,
! [R: partia7496981018696276118t_unit,A2: set_list_a,B: set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A2 @ B )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( A2
= ( zero_s2910681146719230829t_unit @ R ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_151_Ring_Ointegral,axiom,
! [R: partia2175431115845679010xt_a_b,A2: a,B: a] :
( ( field_a_b @ R )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A2 @ B )
= ( zero_a_b @ R ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A2
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_152_Ring_Ointegral,axiom,
! [R: partia2670972154091845814t_unit,A2: list_a,B: list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A2 @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( A2
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_153_Ring_Ointegral,axiom,
! [R: partia2956882679547061052t_unit,A2: list_list_a,B: list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ A2 @ B )
= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( member_list_list_a @ A2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( A2
= ( zero_l347298301471573063t_unit @ R ) )
| ( B
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_154_semiring_Ol__null,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.l_null
thf(fact_155_semiring_Ol__null,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X3 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.l_null
thf(fact_156_semiring_Ol__null,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X3 )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.l_null
thf(fact_157_semiring_Or__null,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X3 @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.r_null
thf(fact_158_semiring_Or__null,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X3 @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.r_null
thf(fact_159_semiring_Or__null,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ X3 @ ( zero_l347298301471573063t_unit @ R ) )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.r_null
thf(fact_160_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_161_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_162_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( one_li8234411390022467901t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_163_ring__irreducible__def,axiom,
( ring_r999134135267193926le_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,A4: a] :
( ( A4
!= ( zero_a_b @ R2 ) )
& ( irredu6211895646901577903xt_a_b @ R2 @ A4 ) ) ) ) ).
% ring_irreducible_def
thf(fact_164_ring__irreducible__def,axiom,
( ring_r932985474545269838t_unit
= ( ^ [R2: partia2670972154091845814t_unit,A4: list_a] :
( ( A4
!= ( zero_l4142658623432671053t_unit @ R2 ) )
& ( irredu4230924414530676029t_unit @ R2 @ A4 ) ) ) ) ).
% ring_irreducible_def
thf(fact_165_ring__irreducible__def,axiom,
( ring_r360171070648044744t_unit
= ( ^ [R2: partia2956882679547061052t_unit,A4: list_list_a] :
( ( A4
!= ( zero_l347298301471573063t_unit @ R2 ) )
& ( irredu4439051761327310013t_unit @ R2 @ A4 ) ) ) ) ).
% ring_irreducible_def
thf(fact_166_principal__domain_Oprimeness__condition,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ P )
= ( ring_ring_prime_a_b @ R @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_167_principal__domain_Oprimeness__condition,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ P )
= ( ring_r6430282645014804837t_unit @ R @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_168_principal__domain_Oprimeness__condition,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( ring_p715737262848045090t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ P )
= ( ring_r5437400583859147359t_unit @ R @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_169_univ__poly__is__principal,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_principal
thf(fact_170_exists__unique__long__division,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ? [X: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ X )
& ! [Y3: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ Y3 )
=> ( Y3 = X ) ) ) ) ) ) ) ).
% exists_unique_long_division
thf(fact_171_x_Oonepideal,axiom,
princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.onepideal
thf(fact_172_dividesI_H,axiom,
! [B: a,G: partia2175431115845679010xt_a_b,A2: a,C: a] :
( ( B
= ( mult_a_ring_ext_a_b @ G @ A2 @ C ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( factor8216151070175719842xt_a_b @ G @ A2 @ B ) ) ) ).
% dividesI'
thf(fact_173_dividesI_H,axiom,
! [B: list_a,G: partia2670972154091845814t_unit,A2: list_a,C: list_a] :
( ( B
= ( mult_l7073676228092353617t_unit @ G @ A2 @ C ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( factor1757716651909850160t_unit @ G @ A2 @ B ) ) ) ).
% dividesI'
thf(fact_174_dividesI_H,axiom,
! [B: list_list_a,G: partia2956882679547061052t_unit,A2: list_list_a,C: list_list_a] :
( ( B
= ( mult_l4853965630390486993t_unit @ G @ A2 @ C ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G ) )
=> ( factor6954119973539764400t_unit @ G @ A2 @ B ) ) ) ).
% dividesI'
thf(fact_175_dividesI_H,axiom,
! [B: a,G: partia8223610829204095565t_unit,A2: a,C: a] :
( ( B
= ( mult_a_Product_unit @ G @ A2 @ C ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( factor3040189038382604065t_unit @ G @ A2 @ B ) ) ) ).
% dividesI'
thf(fact_176_long__division__zero_I1_J,axiom,
! [K: set_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% long_division_zero(1)
thf(fact_177_long__division__closed_I1_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_178_monoid__cancelI,axiom,
( ! [A3: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A3 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B2 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A3 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B2 @ C2 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_179_line__extension__smult__closed,axiom,
! [K: set_a,E: set_a,A2: a,K3: a,U: a] :
( ( subfield_a_b @ K @ r )
=> ( ! [K4: a,V2: a] :
( ( member_a @ K4 @ K )
=> ( ( member_a @ V2 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K4 @ V2 ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K3 @ K )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A2 @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ U ) @ ( embedd971793762689825387on_a_b @ r @ K @ A2 @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_180_a__lcos__mult__one,axiom,
! [M: set_a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M )
= M ) ) ).
% a_lcos_mult_one
thf(fact_181_x_Odivides__mult,axiom,
! [A2: list_a,C: list_a,B: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ).
% x.divides_mult
thf(fact_182_x_Odivides__prod__l,axiom,
! [A2: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ) ).
% x.divides_prod_l
thf(fact_183_x_Odivides__prod__r,axiom,
! [A2: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) ) ) ) ) ).
% x.divides_prod_r
thf(fact_184_x_Odivides__trans,axiom,
! [A2: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ C ) ) ) ) ).
% x.divides_trans
thf(fact_185_x_Om__lcomm,axiom,
! [X3: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Z ) ) ) ) ) ) ).
% x.m_lcomm
thf(fact_186_x_Om__comm,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X3 ) ) ) ) ).
% x.m_comm
thf(fact_187_x_Om__assoc,axiom,
! [X3: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.m_assoc
thf(fact_188_x_Ozero__divides,axiom,
! [A2: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A2 )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.zero_divides
thf(fact_189_x_Osubring__props_I6_J,axiom,
! [K: set_list_a,H1: list_a,H2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H1 @ K )
=> ( ( member_list_a @ H2 @ K )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H2 ) @ K ) ) ) ) ).
% x.subring_props(6)
thf(fact_190_line__extension__in__carrier,axiom,
! [K: set_a,A2: a,E: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K @ A2 @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_191_a__l__coset__subset__G,axiom,
! [H: set_a,X3: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X3 @ H ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_192_x_Odivides__zero,axiom,
! [A2: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.divides_zero
thf(fact_193_x_Osubring__props_I1_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subring_props(1)
thf(fact_194_x_Oone__divides,axiom,
! [A2: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A2 ) ) ).
% x.one_divides
thf(fact_195_x_Oinv__unique,axiom,
! [Y: list_a,X3: list_a,Y2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X3 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% x.inv_unique
thf(fact_196_x_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X )
= X ) )
=> ( U
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.one_unique
thf(fact_197_is__root__poly__mult__imp__is__root,axiom,
! [P: list_a,Q: list_a,X3: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ X3 )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X3 )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X3 ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_198_pprime__iff__pirreducible,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_199_x_Ocring__fieldI2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A3
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ X4 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.cring_fieldI2
thf(fact_200_x_Odivides__mult__lI,axiom,
! [A2: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ).
% x.divides_mult_lI
thf(fact_201_x_Odivides__mult__rI,axiom,
! [A2: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ C ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) ) ) ) ) ) ).
% x.divides_mult_rI
thf(fact_202_x_Odivides__refl,axiom,
! [A2: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ A2 ) ) ).
% x.divides_refl
thf(fact_203_x_Om__closed,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.m_closed
thf(fact_204_x_Or__null,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.r_null
thf(fact_205_x_Ol__null,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.l_null
thf(fact_206_x_Ol__one,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X3 )
= X3 ) ) ).
% x.l_one
thf(fact_207_x_Or__one,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X3 ) ) ).
% x.r_one
thf(fact_208_s_Oring_Ohom__mult,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) @ s2 )
= ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ X3 @ s2 ) @ ( eval_a_b @ r @ Y @ s2 ) ) ) ) ) ).
% s.ring.hom_mult
thf(fact_209_monoid__cancel_Ois__monoid__cancel,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( monoid5798828371819920185xt_a_b @ G ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_210_monoid__cancel_Ois__monoid__cancel,axiom,
! [G: partia2670972154091845814t_unit] :
( ( monoid4303264861975686087t_unit @ G )
=> ( monoid4303264861975686087t_unit @ G ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_211_monoid__cancel_Ois__monoid__cancel,axiom,
! [G: partia8223610829204095565t_unit] :
( ( monoid1999574367301118026t_unit @ G )
=> ( monoid1999574367301118026t_unit @ G ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_212_monoid__cancel_Ol__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,C: a,A2: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ C @ A2 )
= ( mult_a_ring_ext_a_b @ G @ C @ B ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A2 = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_213_monoid__cancel_Ol__cancel,axiom,
! [G: partia2670972154091845814t_unit,C: list_a,A2: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ C @ A2 )
= ( mult_l7073676228092353617t_unit @ G @ C @ B ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( A2 = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_214_monoid__cancel_Ol__cancel,axiom,
! [G: partia2956882679547061052t_unit,C: list_list_a,A2: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( ( mult_l4853965630390486993t_unit @ G @ C @ A2 )
= ( mult_l4853965630390486993t_unit @ G @ C @ B ) )
=> ( ( member_list_list_a @ A2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G ) )
=> ( A2 = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_215_monoid__cancel_Ol__cancel,axiom,
! [G: partia8223610829204095565t_unit,C: a,A2: a,B: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ C @ A2 )
= ( mult_a_Product_unit @ G @ C @ B ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( A2 = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_216_monoid__cancel_Or__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,C: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ A2 @ C )
= ( mult_a_ring_ext_a_b @ G @ B @ C ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A2 = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_217_monoid__cancel_Or__cancel,axiom,
! [G: partia2670972154091845814t_unit,A2: list_a,C: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ A2 @ C )
= ( mult_l7073676228092353617t_unit @ G @ B @ C ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( A2 = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_218_monoid__cancel_Or__cancel,axiom,
! [G: partia2956882679547061052t_unit,A2: list_list_a,C: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( ( mult_l4853965630390486993t_unit @ G @ A2 @ C )
= ( mult_l4853965630390486993t_unit @ G @ B @ C ) )
=> ( ( member_list_list_a @ A2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G ) )
=> ( A2 = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_219_monoid__cancel_Or__cancel,axiom,
! [G: partia8223610829204095565t_unit,A2: a,C: a,B: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( ( mult_a_Product_unit @ G @ A2 @ C )
= ( mult_a_Product_unit @ G @ B @ C ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( A2 = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_220_monoid__cancel_Odivides__mult__l,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ C @ A2 ) @ ( mult_a_ring_ext_a_b @ G @ C @ B ) )
= ( factor8216151070175719842xt_a_b @ G @ A2 @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_221_monoid__cancel_Odivides__mult__l,axiom,
! [G: partia2670972154091845814t_unit,A2: list_a,B: list_a,C: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( factor1757716651909850160t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ C @ A2 ) @ ( mult_l7073676228092353617t_unit @ G @ C @ B ) )
= ( factor1757716651909850160t_unit @ G @ A2 @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_222_monoid__cancel_Odivides__mult__l,axiom,
! [G: partia2956882679547061052t_unit,A2: list_list_a,B: list_list_a,C: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( member_list_list_a @ A2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( factor6954119973539764400t_unit @ G @ ( mult_l4853965630390486993t_unit @ G @ C @ A2 ) @ ( mult_l4853965630390486993t_unit @ G @ C @ B ) )
= ( factor6954119973539764400t_unit @ G @ A2 @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_223_monoid__cancel_Odivides__mult__l,axiom,
! [G: partia8223610829204095565t_unit,A2: a,B: a,C: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ ( mult_a_Product_unit @ G @ C @ A2 ) @ ( mult_a_Product_unit @ G @ C @ B ) )
= ( factor3040189038382604065t_unit @ G @ A2 @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_224_dividesD,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B: a] :
( ( factor8216151070175719842xt_a_b @ G @ A2 @ B )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
& ( B
= ( mult_a_ring_ext_a_b @ G @ A2 @ X ) ) ) ) ).
% dividesD
thf(fact_225_dividesD,axiom,
! [G: partia2670972154091845814t_unit,A2: list_a,B: list_a] :
( ( factor1757716651909850160t_unit @ G @ A2 @ B )
=> ? [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
& ( B
= ( mult_l7073676228092353617t_unit @ G @ A2 @ X ) ) ) ) ).
% dividesD
thf(fact_226_dividesD,axiom,
! [G: partia2956882679547061052t_unit,A2: list_list_a,B: list_list_a] :
( ( factor6954119973539764400t_unit @ G @ A2 @ B )
=> ? [X: list_list_a] :
( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
& ( B
= ( mult_l4853965630390486993t_unit @ G @ A2 @ X ) ) ) ) ).
% dividesD
thf(fact_227_dividesD,axiom,
! [G: partia8223610829204095565t_unit,A2: a,B: a] :
( ( factor3040189038382604065t_unit @ G @ A2 @ B )
=> ? [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
& ( B
= ( mult_a_Product_unit @ G @ A2 @ X ) ) ) ) ).
% dividesD
thf(fact_228_dividesE,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B: a] :
( ( factor8216151070175719842xt_a_b @ G @ A2 @ B )
=> ~ ! [C2: a] :
( ( B
= ( mult_a_ring_ext_a_b @ G @ A2 @ C2 ) )
=> ~ ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% dividesE
thf(fact_229_dividesE,axiom,
! [G: partia2670972154091845814t_unit,A2: list_a,B: list_a] :
( ( factor1757716651909850160t_unit @ G @ A2 @ B )
=> ~ ! [C2: list_a] :
( ( B
= ( mult_l7073676228092353617t_unit @ G @ A2 @ C2 ) )
=> ~ ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ G ) ) ) ) ).
% dividesE
thf(fact_230_dividesE,axiom,
! [G: partia2956882679547061052t_unit,A2: list_list_a,B: list_list_a] :
( ( factor6954119973539764400t_unit @ G @ A2 @ B )
=> ~ ! [C2: list_list_a] :
( ( B
= ( mult_l4853965630390486993t_unit @ G @ A2 @ C2 ) )
=> ~ ( member_list_list_a @ C2 @ ( partia2464479390973590831t_unit @ G ) ) ) ) ).
% dividesE
thf(fact_231_dividesE,axiom,
! [G: partia8223610829204095565t_unit,A2: a,B: a] :
( ( factor3040189038382604065t_unit @ G @ A2 @ B )
=> ~ ! [C2: a] :
( ( B
= ( mult_a_Product_unit @ G @ A2 @ C2 ) )
=> ~ ( member_a @ C2 @ ( partia6735698275553448452t_unit @ G ) ) ) ) ).
% dividesE
thf(fact_232_dividesI,axiom,
! [C: a,G: partia2175431115845679010xt_a_b,B: a,A2: a] :
( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( B
= ( mult_a_ring_ext_a_b @ G @ A2 @ C ) )
=> ( factor8216151070175719842xt_a_b @ G @ A2 @ B ) ) ) ).
% dividesI
thf(fact_233_dividesI,axiom,
! [C: list_a,G: partia2670972154091845814t_unit,B: list_a,A2: list_a] :
( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( B
= ( mult_l7073676228092353617t_unit @ G @ A2 @ C ) )
=> ( factor1757716651909850160t_unit @ G @ A2 @ B ) ) ) ).
% dividesI
thf(fact_234_dividesI,axiom,
! [C: list_list_a,G: partia2956882679547061052t_unit,B: list_list_a,A2: list_list_a] :
( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( B
= ( mult_l4853965630390486993t_unit @ G @ A2 @ C ) )
=> ( factor6954119973539764400t_unit @ G @ A2 @ B ) ) ) ).
% dividesI
thf(fact_235_dividesI,axiom,
! [C: a,G: partia8223610829204095565t_unit,B: a,A2: a] :
( ( member_a @ C @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( B
= ( mult_a_Product_unit @ G @ A2 @ C ) )
=> ( factor3040189038382604065t_unit @ G @ A2 @ B ) ) ) ).
% dividesI
thf(fact_236_factor__def,axiom,
( factor8216151070175719842xt_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b,A4: a,B3: a] :
? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G2 ) )
& ( B3
= ( mult_a_ring_ext_a_b @ G2 @ A4 @ X2 ) ) ) ) ) ).
% factor_def
thf(fact_237_factor__def,axiom,
( factor1757716651909850160t_unit
= ( ^ [G2: partia2670972154091845814t_unit,A4: list_a,B3: list_a] :
? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G2 ) )
& ( B3
= ( mult_l7073676228092353617t_unit @ G2 @ A4 @ X2 ) ) ) ) ) ).
% factor_def
thf(fact_238_factor__def,axiom,
( factor6954119973539764400t_unit
= ( ^ [G2: partia2956882679547061052t_unit,A4: list_list_a,B3: list_list_a] :
? [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G2 ) )
& ( B3
= ( mult_l4853965630390486993t_unit @ G2 @ A4 @ X2 ) ) ) ) ) ).
% factor_def
thf(fact_239_factor__def,axiom,
( factor3040189038382604065t_unit
= ( ^ [G2: partia8223610829204095565t_unit,A4: a,B3: a] :
? [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ G2 ) )
& ( B3
= ( mult_a_Product_unit @ G2 @ A4 @ X2 ) ) ) ) ) ).
% factor_def
thf(fact_240_isgcd__def,axiom,
( isgcd_a_ring_ext_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b,X2: a,A4: a,B3: a] :
( ( factor8216151070175719842xt_a_b @ G2 @ X2 @ A4 )
& ( factor8216151070175719842xt_a_b @ G2 @ X2 @ B3 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( ( factor8216151070175719842xt_a_b @ G2 @ Y4 @ A4 )
& ( factor8216151070175719842xt_a_b @ G2 @ Y4 @ B3 ) )
=> ( factor8216151070175719842xt_a_b @ G2 @ Y4 @ X2 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_241_isgcd__def,axiom,
( isgcd_1118609098697428327t_unit
= ( ^ [G2: partia2670972154091845814t_unit,X2: list_a,A4: list_a,B3: list_a] :
( ( factor1757716651909850160t_unit @ G2 @ X2 @ A4 )
& ( factor1757716651909850160t_unit @ G2 @ X2 @ B3 )
& ! [Y4: list_a] :
( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( ( factor1757716651909850160t_unit @ G2 @ Y4 @ A4 )
& ( factor1757716651909850160t_unit @ G2 @ Y4 @ B3 ) )
=> ( factor1757716651909850160t_unit @ G2 @ Y4 @ X2 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_242_isgcd__def,axiom,
( isgcd_3804025100609598183t_unit
= ( ^ [G2: partia2956882679547061052t_unit,X2: list_list_a,A4: list_list_a,B3: list_list_a] :
( ( factor6954119973539764400t_unit @ G2 @ X2 @ A4 )
& ( factor6954119973539764400t_unit @ G2 @ X2 @ B3 )
& ! [Y4: list_list_a] :
( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( ( factor6954119973539764400t_unit @ G2 @ Y4 @ A4 )
& ( factor6954119973539764400t_unit @ G2 @ Y4 @ B3 ) )
=> ( factor6954119973539764400t_unit @ G2 @ Y4 @ X2 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_243_isgcd__def,axiom,
( isgcd_a_Product_unit
= ( ^ [G2: partia8223610829204095565t_unit,X2: a,A4: a,B3: a] :
( ( factor3040189038382604065t_unit @ G2 @ X2 @ A4 )
& ( factor3040189038382604065t_unit @ G2 @ X2 @ B3 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( ( factor3040189038382604065t_unit @ G2 @ Y4 @ A4 )
& ( factor3040189038382604065t_unit @ G2 @ Y4 @ B3 ) )
=> ( factor3040189038382604065t_unit @ G2 @ Y4 @ X2 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_244_x_Oisgcd__divides__l,axiom,
! [A2: list_a,B: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( isgcd_1118609098697428327t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ A2 @ B ) ) ) ) ).
% x.isgcd_divides_l
thf(fact_245_x_Oisgcd__divides__r,axiom,
! [B: list_a,A2: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A2 )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( isgcd_1118609098697428327t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A2 @ B ) ) ) ) ).
% x.isgcd_divides_r
thf(fact_246_rupture__is__field__iff__pirreducible,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).
% rupture_is_field_iff_pirreducible
thf(fact_247_univ__poly__zero__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ).
% univ_poly_zero_closed
thf(fact_248_univ__poly__zero__closed,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] : ( member_list_list_a @ nil_list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ).
% univ_poly_zero_closed
thf(fact_249_x_Omonoid__cancelI,axiom,
( ! [A3: list_a,B2: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ A3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ B2 ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: list_a,B2: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ C2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B2 @ C2 ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.monoid_cancelI
thf(fact_250_x_Oa__lcos__mult__one,axiom,
! [M: set_list_a] :
( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ M )
= M ) ) ).
% x.a_lcos_mult_one
thf(fact_251_x_Oline__extension__smult__closed,axiom,
! [K: set_list_a,E: set_list_a,A2: list_a,K3: list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [K4: list_a,V2: list_a] :
( ( member_list_a @ K4 @ K )
=> ( ( member_list_a @ V2 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ V2 ) @ E ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K3 @ K )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A2 @ E ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ U ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A2 @ E ) ) ) ) ) ) ) ) ).
% x.line_extension_smult_closed
thf(fact_252_exists__long__division,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ~ ! [B2: list_a] :
( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ! [R4: list_a] :
( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B2 @ R4 ) ) ) ) ) ) ) ) ).
% exists_long_division
thf(fact_253_x_Oa__l__coset__subset__G,axiom,
! [H: set_list_a,X3: list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ H ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.a_l_coset_subset_G
thf(fact_254_poly__add_Ocases,axiom,
! [X3: produc9164743771328383783list_a] :
~ ! [P1: list_a,P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).
% poly_add.cases
thf(fact_255_x_Oline__extension__in__carrier,axiom,
! [K: set_list_a,A2: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A2 @ E ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.line_extension_in_carrier
thf(fact_256_ring_Oeval_Ocong,axiom,
eval_a_b = eval_a_b ).
% ring.eval.cong
thf(fact_257_ring_Oeval_Ocong,axiom,
eval_l34571156754992824t_unit = eval_l34571156754992824t_unit ).
% ring.eval.cong
thf(fact_258_ring_Oconst__term_Ocong,axiom,
const_term_a_b = const_term_a_b ).
% ring.const_term.cong
thf(fact_259_ring_Oconst__term_Ocong,axiom,
const_6738166269504826821t_unit = const_6738166269504826821t_unit ).
% ring.const_term.cong
thf(fact_260_univ__poly__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K ) )
= nil_a ) ).
% univ_poly_zero
thf(fact_261_univ__poly__zero,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) )
= nil_list_a ) ).
% univ_poly_zero
thf(fact_262_long__divisionI,axiom,
! [K: set_a,P: list_a,Q: list_a,B: list_a,R3: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B @ R3 ) )
=> ( ( produc6837034575241423639list_a @ B @ R3 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ) ).
% long_divisionI
thf(fact_263_long__divisionE,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ).
% long_divisionE
thf(fact_264_x_Ocarrier__is__subalgebra,axiom,
! [K: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( embedd1768981623711841426t_unit @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.carrier_is_subalgebra
thf(fact_265_x_Ogenideal__self,axiom,
! [S: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ S @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S ) ) ) ).
% x.genideal_self
thf(fact_266_x_Osubalgebra__in__carrier,axiom,
! [K: set_list_a,V: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ V @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subalgebra_in_carrier
thf(fact_267_x_Osubset__Idl__subset,axiom,
! [I: set_list_a,H: set_list_a] :
( ( ord_le8861187494160871172list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ H @ I )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I ) ) ) ) ).
% x.subset_Idl_subset
thf(fact_268_x_Opoly__of__const__in__carrier,axiom,
! [S2: list_a] :
( ( member_list_a @ S2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.poly_of_const_in_carrier
thf(fact_269_pprimeE_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a,R3: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R3 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
| ( polyno5814909790663948098es_a_b @ r @ P @ R3 ) ) ) ) ) ) ) ) ).
% pprimeE(3)
thf(fact_270_zero__pdivides,axiom,
! [P: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P )
= ( P = nil_a ) ) ).
% zero_pdivides
thf(fact_271_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_272_x_Oconst__term__def,axiom,
! [P: list_list_a] :
( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.const_term_def
thf(fact_273_x_Oeval_Osimps_I1_J,axiom,
( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a )
= ( ^ [Uu: list_a] : ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.eval.simps(1)
thf(fact_274_x_Oconst__term__not__zero,axiom,
! [P: list_list_a] :
( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( P != nil_list_a ) ) ).
% x.const_term_not_zero
thf(fact_275_long__division__closed_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(2)
thf(fact_276_x_Oeval__poly__of__const,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) @ Y )
= X3 ) ) ).
% x.eval_poly_of_const
thf(fact_277_long__division__zero_I2_J,axiom,
! [K: set_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% long_division_zero(2)
thf(fact_278_x_Oeval__in__carrier__2,axiom,
! [X3: list_list_a,Y: list_a] :
( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.eval_in_carrier_2
thf(fact_279_pdivides__iff__shell,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) ) ) ) ) ).
% pdivides_iff_shell
thf(fact_280_pdivides__imp__root__sharing,axiom,
! [P: list_a,Q: list_a,A2: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( eval_a_b @ r @ P @ A2 )
= ( zero_a_b @ r ) )
=> ( ( eval_a_b @ r @ Q @ A2 )
= ( zero_a_b @ r ) ) ) ) ) ) ).
% pdivides_imp_root_sharing
thf(fact_281_pmod__zero__iff__pdivides,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ P @ Q )
= nil_a )
= ( polyno5814909790663948098es_a_b @ r @ Q @ P ) ) ) ) ) ).
% pmod_zero_iff_pdivides
thf(fact_282_ring_Opoly__of__const_Ocong,axiom,
poly_o8716471131768098070t_unit = poly_o8716471131768098070t_unit ).
% ring.poly_of_const.cong
thf(fact_283_ring_Opoly__of__const_Ocong,axiom,
poly_of_const_a_b = poly_of_const_a_b ).
% ring.poly_of_const.cong
thf(fact_284_pdivides__imp__splitted,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ) ) ) ).
% pdivides_imp_splitted
thf(fact_285_x_Ois__root__def,axiom,
! [P: list_list_a,X3: list_a] :
( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X3 )
= ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( P != nil_list_a ) ) ) ).
% x.is_root_def
thf(fact_286_x_Oeval__var,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X3 )
= X3 ) ) ).
% x.eval_var
thf(fact_287_same__pmod__iff__pdivides,axiom,
! [K: set_a,A2: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A2 @ Q )
= ( polynomial_pmod_a_b @ r @ B @ Q ) )
= ( polyno5814909790663948098es_a_b @ r @ Q @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ A2 @ B ) ) ) ) ) ) ) ).
% same_pmod_iff_pdivides
thf(fact_288_pdiv__pmod,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% pdiv_pmod
thf(fact_289_x_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K: set_list_a,E: set_list_a,V: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
=> ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ V @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ V ) ) ) ) ) ).
% x.subalbegra_incl_imp_finite_dimension
thf(fact_290_pprimeI,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ R4 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q2 )
| ( polyno5814909790663948098es_a_b @ r @ P @ R4 ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ) ) ).
% pprimeI
thf(fact_291_x_Ofinite__dimension__imp__subalgebra,axiom,
! [K: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
=> ( embedd1768981623711841426t_unit @ K @ E @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.finite_dimension_imp_subalgebra
thf(fact_292_eval__poly__of__const,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_of_const_a_b @ r @ X3 ) @ Y )
= X3 ) ) ).
% eval_poly_of_const
thf(fact_293_poly__of__const__in__carrier,axiom,
! [S2: a] :
( ( member_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( poly_of_const_a_b @ r @ S2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% poly_of_const_in_carrier
thf(fact_294_x_Oadd_Ol__cancel,axiom,
! [C: list_a,A2: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A2 = B ) ) ) ) ) ).
% x.add.l_cancel
thf(fact_295_x_Oadd_Om__assoc,axiom,
! [X3: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.add.m_assoc
thf(fact_296_x_Oadd_Om__comm,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X3 ) ) ) ) ).
% x.add.m_comm
thf(fact_297_x_Oadd_Om__lcomm,axiom,
! [X3: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Z ) ) ) ) ) ) ).
% x.add.m_lcomm
thf(fact_298_x_Oadd_Or__cancel,axiom,
! [A2: list_a,C: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ C )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A2 = B ) ) ) ) ) ).
% x.add.r_cancel
thf(fact_299_x_Osubring__props_I7_J,axiom,
! [K: set_list_a,H1: list_a,H2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H1 @ K )
=> ( ( member_list_a @ H2 @ K )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H2 ) @ K ) ) ) ) ).
% x.subring_props(7)
thf(fact_300_x_OUnits__closed,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_closed
thf(fact_301_x_Otelescopic__base__dim_I1_J,axiom,
! [K: set_list_a,F2: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subfie1779122896746047282t_unit @ F2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ F2 )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F2 @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E ) ) ) ) ) ).
% x.telescopic_base_dim(1)
thf(fact_302_x_Oadd_Oinv__comm,axiom,
! [X3: list_a,Y: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.inv_comm
thf(fact_303_x_Oadd_Ol__inv__ex,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.l_inv_ex
thf(fact_304_x_Oadd_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X )
= X ) )
=> ( U
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.one_unique
thf(fact_305_x_Oadd_Or__inv__ex,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.r_inv_ex
thf(fact_306_x_Ominus__unique,axiom,
! [Y: list_a,X3: list_a,Y2: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% x.minus_unique
thf(fact_307_x_Ol__distr,axiom,
! [X3: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Z ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.l_distr
thf(fact_308_x_Or__distr,axiom,
! [X3: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X3 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ Y ) ) ) ) ) ) ).
% x.r_distr
thf(fact_309_x_Oprod__unit__l,axiom,
! [A2: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.prod_unit_l
thf(fact_310_x_Oprod__unit__r,axiom,
! [A2: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.prod_unit_r
thf(fact_311_x_Ounit__factor,axiom,
! [A2: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.unit_factor
thf(fact_312_x_Oirreducible__prod__lI,axiom,
! [B: list_a,A2: list_a] :
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B )
=> ( ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) ) ) ) ) ) ).
% x.irreducible_prod_lI
thf(fact_313_x_Oirreducible__prod__rI,axiom,
! [A2: list_a,B: list_a] :
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) ) ) ) ) ) ).
% x.irreducible_prod_rI
thf(fact_314_x_Ounit__divides,axiom,
! [U: list_a,A2: list_a] :
( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ A2 ) ) ) ).
% x.unit_divides
thf(fact_315_x_Odivides__unit,axiom,
! [A2: list_a,U: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ U )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.divides_unit
thf(fact_316_x_OUnits__inv__comm,axiom,
! [X3: list_a,Y: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X3 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.Units_inv_comm
thf(fact_317_x_Oline__extension__mem__iff,axiom,
! [U: list_a,K: set_list_a,A2: list_a,E: set_list_a] :
( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A2 @ E ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ K )
& ? [Y4: list_a] :
( ( member_list_a @ Y4 @ E )
& ( U
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ A2 ) @ Y4 ) ) ) ) ) ) ).
% x.line_extension_mem_iff
thf(fact_318_long__division__add_I2_J,axiom,
! [K: set_a,A2: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A2 @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ A2 @ Q ) @ ( polynomial_pmod_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(2)
thf(fact_319_long__division__add__iff,axiom,
! [K: set_a,A2: list_a,B: list_a,C: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A2 @ Q )
= ( polynomial_pmod_a_b @ r @ B @ Q ) )
= ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A2 @ C ) @ Q )
= ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ).
% long_division_add_iff
thf(fact_320_long__division__add_I1_J,axiom,
! [K: set_a,A2: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A2 @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ A2 @ Q ) @ ( polynomial_pdiv_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(1)
thf(fact_321_pprimeE_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pprimeE(2)
thf(fact_322_x_OUnits__r__inv__ex,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Units_r_inv_ex
thf(fact_323_x_OUnits__l__inv__ex,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X3 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Units_l_inv_ex
thf(fact_324_x_Oa__lcos__m__assoc,axiom,
! [M: set_list_a,G3: list_a,H3: list_a] :
( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ G3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ H3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G3 @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H3 @ M ) )
= ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G3 @ H3 ) @ M ) ) ) ) ) ).
% x.a_lcos_m_assoc
thf(fact_325_x_Odivides__one,axiom,
! [A2: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.divides_one
thf(fact_326_x_OUnit__eq__dividesone,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Unit_eq_dividesone
thf(fact_327_x_Oadd_Om__closed,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.m_closed
thf(fact_328_x_Oadd_Oright__cancel,axiom,
! [X3: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X3 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X3 ) )
= ( Y = Z ) ) ) ) ) ).
% x.add.right_cancel
thf(fact_329_x_Ofinite__ring__finite__units,axiom,
( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( finite_finite_list_a @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.finite_ring_finite_units
thf(fact_330_x_OUnits__m__closed,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Units_m_closed
thf(fact_331_x_OUnits__one__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.Units_one_closed
thf(fact_332_x_Ominus__closed,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.minus_closed
thf(fact_333_x_Oadd_Ol__cancel__one,axiom,
! [X3: list_a,A2: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ A2 )
= X3 )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one
thf(fact_334_x_Oadd_Ol__cancel__one_H,axiom,
! [X3: list_a,A2: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X3
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ A2 ) )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one'
thf(fact_335_x_Oadd_Or__cancel__one,axiom,
! [X3: list_a,A2: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ X3 )
= X3 )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one
thf(fact_336_x_Oadd_Or__cancel__one_H,axiom,
! [X3: list_a,A2: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X3
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ X3 ) )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one'
thf(fact_337_x_Ol__zero,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X3 )
= X3 ) ) ).
% x.l_zero
thf(fact_338_x_Or__zero,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X3 ) ) ).
% x.r_zero
thf(fact_339_x_OUnits__l__cancel,axiom,
! [X3: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% x.Units_l_cancel
thf(fact_340_x_Or__right__minus__eq,axiom,
! [A2: list_a,B: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( A2 = B ) ) ) ) ).
% x.r_right_minus_eq
thf(fact_341_s_Ohom__sub,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) @ s2 )
= ( a_minus_a_b @ r @ ( eval_a_b @ r @ X3 @ s2 ) @ ( eval_a_b @ r @ Y @ s2 ) ) ) ) ) ).
% s.hom_sub
thf(fact_342_abelian__monoid_Oa__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( add_a_b @ G @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_343_abelian__monoid_Oa__closed,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ G @ X3 @ Y ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_344_abelian__monoid_Oa__closed,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a,Y: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ G @ X3 @ Y ) @ ( partia2464479390973590831t_unit @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_345_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X3 @ ( add_a_b @ G @ Y @ Z ) )
= ( add_a_b @ G @ Y @ ( add_a_b @ G @ X3 @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_346_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y: list_a,Z: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X3 @ ( add_li7652885771158616974t_unit @ G @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ G @ Y @ ( add_li7652885771158616974t_unit @ G @ X3 @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_347_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ X3 @ ( add_li174743652000525320t_unit @ G @ Y @ Z ) )
= ( add_li174743652000525320t_unit @ G @ Y @ ( add_li174743652000525320t_unit @ G @ X3 @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_348_abelian__monoid_Oa__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( add_a_b @ G @ X3 @ Y ) @ Z )
= ( add_a_b @ G @ X3 @ ( add_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_349_abelian__monoid_Oa__assoc,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y: list_a,Z: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( add_li7652885771158616974t_unit @ G @ X3 @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ G @ X3 @ ( add_li7652885771158616974t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_350_abelian__monoid_Oa__assoc,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ ( add_li174743652000525320t_unit @ G @ X3 @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ G @ X3 @ ( add_li174743652000525320t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_351_abelian__monoid_Oa__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X3 @ Y )
= ( add_a_b @ G @ Y @ X3 ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_352_abelian__monoid_Oa__comm,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X3 @ Y )
= ( add_li7652885771158616974t_unit @ G @ Y @ X3 ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_353_abelian__monoid_Oa__comm,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a,Y: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ X3 @ Y )
= ( add_li174743652000525320t_unit @ G @ Y @ X3 ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_354_abelian__monoidE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_355_abelian__monoidE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_356_abelian__monoidE_I1_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y: list_list_a] :
( ( abelia3641329199688042803t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X3 @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_357_abelian__monoidE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X3 @ Y ) @ Z )
= ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_358_abelian__monoidE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y: list_a,Z: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ X3 @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_359_abelian__monoidE_I3_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( abelia3641329199688042803t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X3 @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ R @ X3 @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_360_abelian__monoidE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ Y )
= ( add_a_b @ R @ Y @ X3 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_361_abelian__monoidE_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X3 @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X3 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_362_abelian__monoidE_I5_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y: list_list_a] :
( ( abelia3641329199688042803t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X3 @ Y )
= ( add_li174743652000525320t_unit @ R @ Y @ X3 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_363_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_364_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_365_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X3 @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_366_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X3 @ Y ) @ Z )
= ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_367_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ X3 @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_368_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X3 @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ R @ X3 @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_369_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ Y )
= ( add_a_b @ R @ Y @ X3 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_370_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X3 @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X3 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_371_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X3 @ Y )
= ( add_li174743652000525320t_unit @ R @ Y @ X3 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_372_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X3 @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_373_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X3 @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ R @ Y @ ( add_li7652885771158616974t_unit @ R @ X3 @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_374_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X3 @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) )
= ( add_li174743652000525320t_unit @ R @ Y @ ( add_li174743652000525320t_unit @ R @ X3 @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_375_abelian__monoidE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% abelian_monoidE(4)
thf(fact_376_abelian__monoidE_I4_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% abelian_monoidE(4)
thf(fact_377_abelian__monoidE_I4_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( abelia3641329199688042803t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% abelian_monoidE(4)
thf(fact_378_abelian__monoid_Ol__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( zero_a_b @ G ) @ X3 )
= X3 ) ) ) ).
% abelian_monoid.l_zero
thf(fact_379_abelian__monoid_Ol__zero,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( zero_l4142658623432671053t_unit @ G ) @ X3 )
= X3 ) ) ) ).
% abelian_monoid.l_zero
thf(fact_380_abelian__monoid_Ol__zero,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ ( zero_l347298301471573063t_unit @ G ) @ X3 )
= X3 ) ) ) ).
% abelian_monoid.l_zero
thf(fact_381_abelian__monoid_Or__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X3 @ ( zero_a_b @ G ) )
= X3 ) ) ) ).
% abelian_monoid.r_zero
thf(fact_382_abelian__monoid_Or__zero,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X3 @ ( zero_l4142658623432671053t_unit @ G ) )
= X3 ) ) ) ).
% abelian_monoid.r_zero
thf(fact_383_abelian__monoid_Or__zero,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ X3 @ ( zero_l347298301471573063t_unit @ G ) )
= X3 ) ) ) ).
% abelian_monoid.r_zero
thf(fact_384_abelian__monoid_Ominus__unique,axiom,
! [G: partia2175431115845679010xt_a_b,Y: a,X3: a,Y2: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( ( add_a_b @ G @ Y @ X3 )
= ( zero_a_b @ G ) )
=> ( ( ( add_a_b @ G @ X3 @ Y2 )
= ( zero_a_b @ G ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_385_abelian__monoid_Ominus__unique,axiom,
! [G: partia2670972154091845814t_unit,Y: list_a,X3: list_a,Y2: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( ( add_li7652885771158616974t_unit @ G @ Y @ X3 )
= ( zero_l4142658623432671053t_unit @ G ) )
=> ( ( ( add_li7652885771158616974t_unit @ G @ X3 @ Y2 )
= ( zero_l4142658623432671053t_unit @ G ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_386_abelian__monoid_Ominus__unique,axiom,
! [G: partia2956882679547061052t_unit,Y: list_list_a,X3: list_list_a,Y2: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( ( add_li174743652000525320t_unit @ G @ Y @ X3 )
= ( zero_l347298301471573063t_unit @ G ) )
=> ( ( ( add_li174743652000525320t_unit @ G @ X3 @ Y2 )
= ( zero_l347298301471573063t_unit @ G ) )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_387_abelian__monoidI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X: a,Y5: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y5 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ! [X: a,Y5: a,Z2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y5 ) @ Z2 )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y5 @ Z2 ) ) ) ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) )
=> ( ! [X: a,Y5: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y5 )
= ( add_a_b @ R @ Y5 @ X ) ) ) )
=> ( abelian_monoid_a_b @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_388_abelian__monoidI,axiom,
! [R: partia2670972154091845814t_unit] :
( ! [X: list_a,Y5: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y5 ) @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ! [X: list_a,Y5: list_a,Z2: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y5 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y5 @ Z2 ) ) ) ) ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) )
=> ( ! [X: list_a,Y5: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ Y5 )
= ( add_li7652885771158616974t_unit @ R @ Y5 @ X ) ) ) )
=> ( abelia226231641709521465t_unit @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_389_abelian__monoidI,axiom,
! [R: partia2956882679547061052t_unit] :
( ! [X: list_list_a,Y5: list_list_a] :
( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X @ Y5 ) @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ! [X: list_list_a,Y5: list_list_a,Z2: list_list_a] :
( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y5 ) @ Z2 )
= ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y5 @ Z2 ) ) ) ) ) )
=> ( ! [X: list_list_a] :
( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
= X ) )
=> ( ! [X: list_list_a,Y5: list_list_a] :
( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X @ Y5 )
= ( add_li174743652000525320t_unit @ R @ Y5 @ X ) ) ) )
=> ( abelia3641329199688042803t_unit @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_390_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ ( zero_a_b @ R ) )
= X3 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_391_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X3 @ ( zero_l4142658623432671053t_unit @ R ) )
= X3 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_392_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X3 @ ( zero_l347298301471573063t_unit @ R ) )
= X3 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_393_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_394_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_395_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_396_semiring_Or__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X3 @ Y ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X3 ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_397_semiring_Or__distr,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ Z @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y ) )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z @ X3 ) @ ( mult_l7073676228092353617t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_398_semiring_Or__distr,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ Z @ ( add_li174743652000525320t_unit @ R @ X3 @ Y ) )
= ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ Z @ X3 ) @ ( mult_l4853965630390486993t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_399_semiring_Ol__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X3 @ Y ) @ Z )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_400_semiring_Ol__distr,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X3 @ Z ) @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_401_semiring_Ol__distr,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X3 @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X3 @ Z ) @ ( mult_l4853965630390486993t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_402_primeE,axiom,
! [G: partia2175431115845679010xt_a_b,P: a] :
( ( prime_a_ring_ext_a_b @ G @ P )
=> ~ ( ~ ( member_a @ P @ ( units_a_ring_ext_a_b @ G ) )
=> ~ ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ G ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ P @ ( mult_a_ring_ext_a_b @ G @ X4 @ Xa ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ P @ X4 )
| ( factor8216151070175719842xt_a_b @ G @ P @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_403_primeE,axiom,
! [G: partia2670972154091845814t_unit,P: list_a] :
( ( prime_2011924034616061926t_unit @ G @ P )
=> ~ ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ G ) )
=> ~ ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ G ) )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( factor1757716651909850160t_unit @ G @ P @ ( mult_l7073676228092353617t_unit @ G @ X4 @ Xa ) )
=> ( ( factor1757716651909850160t_unit @ G @ P @ X4 )
| ( factor1757716651909850160t_unit @ G @ P @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_404_primeE,axiom,
! [G: partia2956882679547061052t_unit,P: list_list_a] :
( ( prime_1232919612140715622t_unit @ G @ P )
=> ~ ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ G ) )
=> ~ ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ ( partia2464479390973590831t_unit @ G ) )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( factor6954119973539764400t_unit @ G @ P @ ( mult_l4853965630390486993t_unit @ G @ X4 @ Xa ) )
=> ( ( factor6954119973539764400t_unit @ G @ P @ X4 )
| ( factor6954119973539764400t_unit @ G @ P @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_405_primeE,axiom,
! [G: partia8223610829204095565t_unit,P: a] :
( ( prime_a_Product_unit @ G @ P )
=> ~ ( ~ ( member_a @ P @ ( units_a_Product_unit @ G ) )
=> ~ ! [X4: a] :
( ( member_a @ X4 @ ( partia6735698275553448452t_unit @ G ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ P @ ( mult_a_Product_unit @ G @ X4 @ Xa ) )
=> ( ( factor3040189038382604065t_unit @ G @ P @ X4 )
| ( factor3040189038382604065t_unit @ G @ P @ Xa ) ) ) ) ) ) ) ).
% primeE
thf(fact_406_primeI,axiom,
! [P: a,G: partia2175431115845679010xt_a_b] :
( ~ ( member_a @ P @ ( units_a_ring_ext_a_b @ G ) )
=> ( ! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ P @ ( mult_a_ring_ext_a_b @ G @ A3 @ B2 ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ P @ A3 )
| ( factor8216151070175719842xt_a_b @ G @ P @ B2 ) ) ) ) )
=> ( prime_a_ring_ext_a_b @ G @ P ) ) ) ).
% primeI
thf(fact_407_primeI,axiom,
! [P: list_a,G: partia2670972154091845814t_unit] :
( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ G ) )
=> ( ! [A3: list_a,B2: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( factor1757716651909850160t_unit @ G @ P @ ( mult_l7073676228092353617t_unit @ G @ A3 @ B2 ) )
=> ( ( factor1757716651909850160t_unit @ G @ P @ A3 )
| ( factor1757716651909850160t_unit @ G @ P @ B2 ) ) ) ) )
=> ( prime_2011924034616061926t_unit @ G @ P ) ) ) ).
% primeI
thf(fact_408_primeI,axiom,
! [P: list_list_a,G: partia2956882679547061052t_unit] :
( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ G ) )
=> ( ! [A3: list_list_a,B2: list_list_a] :
( ( member_list_list_a @ A3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ B2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( factor6954119973539764400t_unit @ G @ P @ ( mult_l4853965630390486993t_unit @ G @ A3 @ B2 ) )
=> ( ( factor6954119973539764400t_unit @ G @ P @ A3 )
| ( factor6954119973539764400t_unit @ G @ P @ B2 ) ) ) ) )
=> ( prime_1232919612140715622t_unit @ G @ P ) ) ) ).
% primeI
thf(fact_409_primeI,axiom,
! [P: a,G: partia8223610829204095565t_unit] :
( ~ ( member_a @ P @ ( units_a_Product_unit @ G ) )
=> ( ! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ P @ ( mult_a_Product_unit @ G @ A3 @ B2 ) )
=> ( ( factor3040189038382604065t_unit @ G @ P @ A3 )
| ( factor3040189038382604065t_unit @ G @ P @ B2 ) ) ) ) )
=> ( prime_a_Product_unit @ G @ P ) ) ) ).
% primeI
thf(fact_410_prime__def,axiom,
( prime_a_ring_ext_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b,P3: a] :
( ~ ( member_a @ P3 @ ( units_a_ring_ext_a_b @ G2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ P3 @ ( mult_a_ring_ext_a_b @ G2 @ X2 @ Y4 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ P3 @ X2 )
| ( factor8216151070175719842xt_a_b @ G2 @ P3 @ Y4 ) ) ) ) ) ) ) ) ).
% prime_def
thf(fact_411_prime__def,axiom,
( prime_2011924034616061926t_unit
= ( ^ [G2: partia2670972154091845814t_unit,P3: list_a] :
( ~ ( member_list_a @ P3 @ ( units_2932844235741507942t_unit @ G2 ) )
& ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ! [Y4: list_a] :
( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( factor1757716651909850160t_unit @ G2 @ P3 @ ( mult_l7073676228092353617t_unit @ G2 @ X2 @ Y4 ) )
=> ( ( factor1757716651909850160t_unit @ G2 @ P3 @ X2 )
| ( factor1757716651909850160t_unit @ G2 @ P3 @ Y4 ) ) ) ) ) ) ) ) ).
% prime_def
thf(fact_412_prime__def,axiom,
( prime_1232919612140715622t_unit
= ( ^ [G2: partia2956882679547061052t_unit,P3: list_list_a] :
( ~ ( member_list_list_a @ P3 @ ( units_4903515905731149798t_unit @ G2 ) )
& ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ! [Y4: list_list_a] :
( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( factor6954119973539764400t_unit @ G2 @ P3 @ ( mult_l4853965630390486993t_unit @ G2 @ X2 @ Y4 ) )
=> ( ( factor6954119973539764400t_unit @ G2 @ P3 @ X2 )
| ( factor6954119973539764400t_unit @ G2 @ P3 @ Y4 ) ) ) ) ) ) ) ) ).
% prime_def
thf(fact_413_prime__def,axiom,
( prime_a_Product_unit
= ( ^ [G2: partia8223610829204095565t_unit,P3: a] :
( ~ ( member_a @ P3 @ ( units_a_Product_unit @ G2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( factor3040189038382604065t_unit @ G2 @ P3 @ ( mult_a_Product_unit @ G2 @ X2 @ Y4 ) )
=> ( ( factor3040189038382604065t_unit @ G2 @ P3 @ X2 )
| ( factor3040189038382604065t_unit @ G2 @ P3 @ Y4 ) ) ) ) ) ) ) ) ).
% prime_def
thf(fact_414_field_Opdivides__imp__splitted,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,Q: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( Q != nil_set_list_a )
=> ( ( polyno7858167711734664505t_unit @ R @ Q )
=> ( ( polyno9075941895896075626t_unit @ R @ P @ Q )
=> ( polyno7858167711734664505t_unit @ R @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_415_field_Opdivides__imp__splitted,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( Q != nil_list_list_a )
=> ( ( polyno5970451904377802771t_unit @ R @ Q )
=> ( ( polyno4453881341673752516t_unit @ R @ P @ Q )
=> ( polyno5970451904377802771t_unit @ R @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_416_field_Opdivides__imp__splitted,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno6259083269128200473t_unit @ R @ Q )
=> ( ( polyno8016796738000020810t_unit @ R @ P @ Q )
=> ( polyno6259083269128200473t_unit @ R @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_417_field_Opdivides__imp__splitted,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ R @ Q )
=> ( ( polyno5814909790663948098es_a_b @ R @ P @ Q )
=> ( polyno8329700637149614481ed_a_b @ R @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_418_map__in__poly__ring__carrier,axiom,
! [P: list_a,F: a > list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [A3: a] :
( ( A3
!= ( zero_a_b @ r ) )
=> ( ( F @ A3 )
!= nil_a ) )
=> ( member_list_list_a @ ( map_a_list_a @ F @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).
% map_in_poly_ring_carrier
thf(fact_419_pirreducibleI,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ R4 ) )
=> ( ( member_list_a @ Q2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( member_list_a @ R4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ) ) ).
% pirreducibleI
thf(fact_420_x_Oconst__term__simprules_I1_J,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.const_term_simprules(1)
thf(fact_421_pirreducible__pow__pdivides__iff,axiom,
! [K: set_a,P: list_a,Q: list_a,R3: list_a,N: nat] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ~ ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R3 ) )
= ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) @ R3 ) ) ) ) ) ) ) ) ).
% pirreducible_pow_pdivides_iff
thf(fact_422_x_Oeval__in__carrier,axiom,
! [P: list_list_a,X3: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.eval_in_carrier
thf(fact_423_x_Oadd__pow__ldistr__int,axiom,
! [A2: list_a,B: list_a,K3: int] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ A2 ) @ B )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) ) ) ) ) ).
% x.add_pow_ldistr_int
thf(fact_424_x_Oadd__pow__rdistr__int,axiom,
! [A2: list_a,B: list_a,K3: int] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ B ) )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) ) ) ) ) ).
% x.add_pow_rdistr_int
thf(fact_425_Units__closed,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_426_add_Ol__cancel,axiom,
! [C: a,A2: a,B: a] :
( ( ( add_a_b @ r @ C @ A2 )
= ( add_a_b @ r @ C @ B ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B ) ) ) ) ) ).
% add.l_cancel
thf(fact_427_add_Or__cancel,axiom,
! [A2: a,C: a,B: a] :
( ( ( add_a_b @ r @ A2 @ C )
= ( add_a_b @ r @ B @ C ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B ) ) ) ) ) ).
% add.r_cancel
thf(fact_428_a__assoc,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X3 @ Y ) @ Z )
= ( add_a_b @ r @ X3 @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_429_a__comm,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ Y )
= ( add_a_b @ r @ Y @ X3 ) ) ) ) ).
% a_comm
thf(fact_430_a__lcomm,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X3 @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_431_subring__props_I7_J,axiom,
! [K: set_a,H1: a,H2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H2 @ K )
=> ( member_a @ ( add_a_b @ r @ H1 @ H2 ) @ K ) ) ) ) ).
% subring_props(7)
thf(fact_432_add_Oinv__comm,axiom,
! [X3: a,Y: a] :
( ( ( add_a_b @ r @ X3 @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X3 )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_433_add_Ol__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X3 )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_434_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X )
= X ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_435_add_Or__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X3 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_436_local_Ominus__unique,axiom,
! [Y: a,X3: a,Y2: a] :
( ( ( add_a_b @ r @ Y @ X3 )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X3 @ Y2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_437_prod__unit__l,axiom,
! [A2: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_438_prod__unit__r,axiom,
! [A2: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_439_unit__factor,axiom,
! [A2: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_440_l__distr,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X3 @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_441_r__distr,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X3 @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X3 ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_442_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_443_divides__unit,axiom,
! [A2: a,U: a] :
( ( factor8216151070175719842xt_a_b @ r @ A2 @ U )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% divides_unit
thf(fact_444_unit__divides,axiom,
! [U: a,A2: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ U @ A2 ) ) ) ).
% unit_divides
thf(fact_445_Units__inv__comm,axiom,
! [X3: a,Y: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_inv_comm
thf(fact_446_line__extension__mem__iff,axiom,
! [U: a,K: set_a,A2: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A2 @ E ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ K )
& ? [Y4: a] :
( ( member_a @ Y4 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ A2 ) @ Y4 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_447_ring__irreducibleE_I4_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_448_Units__r__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_449_Units__l__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_450_divides__one,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A2 @ ( one_a_ring_ext_a_b @ r ) )
= ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% divides_one
thf(fact_451_Unit__eq__dividesone,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
= ( factor8216151070175719842xt_a_b @ r @ U @ ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Unit_eq_dividesone
thf(fact_452_a__lcos__m__assoc,axiom,
! [M: set_a,G3: a,H3: a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G3 @ ( a_l_coset_a_b @ r @ H3 @ M ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G3 @ H3 ) @ M ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_453_irreducible__prod__rI,axiom,
! [A2: a,B: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A2 )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) ) ) ) ) ) ).
% irreducible_prod_rI
thf(fact_454_irreducible__prod__lI,axiom,
! [B: a,A2: a] :
( ( irredu6211895646901577903xt_a_b @ r @ B )
=> ( ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) ) ) ) ) ) ).
% irreducible_prod_lI
thf(fact_455_ring__irreducibleE_I5_J,axiom,
! [R3: a,A2: a,B: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R3
= ( mult_a_ring_ext_a_b @ r @ A2 @ B ) )
=> ( ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_456_univ__poly__not__field,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ~ ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_not_field
thf(fact_457_x_OUnits__pow__closed,axiom,
! [X3: list_a,D: nat] :
( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ D ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_pow_closed
thf(fact_458_univ__poly__is__abelian__monoid,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( abelia226231641709521465t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_abelian_monoid
thf(fact_459_const__term__simprules__shell_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_a @ ( const_term_a_b @ r @ P ) @ K ) ) ) ).
% const_term_simprules_shell(1)
thf(fact_460_x_Oadd_Oint__pow__mult__distrib,axiom,
! [X3: list_a,Y: list_a,I2: int] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X3 ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ X3 ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ Y ) ) ) ) ) ) ).
% x.add.int_pow_mult_distrib
thf(fact_461_x_Oadd_Oint__pow__distrib,axiom,
! [X3: list_a,Y: list_a,I2: int] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ X3 ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ Y ) ) ) ) ) ).
% x.add.int_pow_distrib
thf(fact_462_polynomial__pow__not__zero,axiom,
! [P: list_a,N: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N )
!= nil_a ) ) ) ).
% polynomial_pow_not_zero
thf(fact_463_x_Opow__mult__distrib,axiom,
! [X3: list_a,Y: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X3 ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ N ) ) ) ) ) ) ).
% x.pow_mult_distrib
thf(fact_464_x_Onat__pow__distrib,axiom,
! [X3: list_a,Y: list_a,N: nat] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ N ) ) ) ) ) ).
% x.nat_pow_distrib
thf(fact_465_x_Onat__pow__comm,axiom,
! [X3: list_a,N: nat,M2: nat] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ M2 ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ M2 ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ N ) ) ) ) ).
% x.nat_pow_comm
thf(fact_466_x_Ogroup__commutes__pow,axiom,
! [X3: list_a,Y: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X3 ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ N ) @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ N ) ) ) ) ) ) ).
% x.group_commutes_pow
thf(fact_467_subring__polynomial__pow__not__zero,axiom,
! [K: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N )
!= nil_a ) ) ) ) ).
% subring_polynomial_pow_not_zero
thf(fact_468_const__term__simprules__shell_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) )
= ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ) ).
% const_term_simprules_shell(3)
thf(fact_469_pdivides__zero,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P @ nil_a ) ) ) ).
% pdivides_zero
thf(fact_470_univ__poly__a__minus__consistent,axiom,
! [K: set_a,Q: list_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) ) ) ) ).
% univ_poly_a_minus_consistent
thf(fact_471_pirreducibleE_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( P != nil_a ) ) ) ) ).
% pirreducibleE(1)
thf(fact_472_pirreducibleE_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pirreducibleE(2)
thf(fact_473_const__term__simprules__shell_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ) ).
% const_term_simprules_shell(2)
thf(fact_474_pirreducibleE_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a,R3: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R3 ) )
=> ( ( member_list_a @ Q @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ) ) ) ).
% pirreducibleE(3)
thf(fact_475_local_Oadd_Oright__cancel,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X3 )
= ( add_a_b @ r @ Z @ X3 ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_476_a__closed,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X3 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_477_map__norm__in__poly__ring__carrier,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_list_a @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ).
% map_norm_in_poly_ring_carrier
thf(fact_478_Units__m__closed,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_479_Units__one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_one_closed
thf(fact_480_add_Ol__cancel__one,axiom,
! [X3: a,A2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X3 @ A2 )
= X3 )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_481_add_Ol__cancel__one_H,axiom,
! [X3: a,A2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X3
= ( add_a_b @ r @ X3 @ A2 ) )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_482_add_Or__cancel__one,axiom,
! [X3: a,A2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A2 @ X3 )
= X3 )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_483_add_Or__cancel__one_H,axiom,
! [X3: a,A2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X3
= ( add_a_b @ r @ A2 @ X3 ) )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_484_l__zero,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X3 )
= X3 ) ) ).
% l_zero
thf(fact_485_r__zero,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( zero_a_b @ r ) )
= X3 ) ) ).
% r_zero
thf(fact_486_Units__l__cancel,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y )
= ( mult_a_ring_ext_a_b @ r @ X3 @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% Units_l_cancel
thf(fact_487_finite__ring__finite__units,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( finite_finite_a @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% finite_ring_finite_units
thf(fact_488_carrier__polynomial__shell,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% carrier_polynomial_shell
thf(fact_489_x_Onat__pow__closed,axiom,
! [X3: list_a,N: nat] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.nat_pow_closed
thf(fact_490_x_Oadd_Oint__pow__closed,axiom,
! [X3: list_a,I2: int] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ X3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.int_pow_closed
thf(fact_491_x_Oadd_Oint__pow__one,axiom,
! [Z: int] :
( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.add.int_pow_one
thf(fact_492_x_Onat__pow__one,axiom,
! [N: nat] :
( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.nat_pow_one
thf(fact_493_s_Oring_Ohom__add,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) @ s2 )
= ( add_a_b @ r @ ( eval_a_b @ r @ X3 @ s2 ) @ ( eval_a_b @ r @ Y @ s2 ) ) ) ) ) ).
% s.ring.hom_add
thf(fact_494_subringE_I2_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H ) ) ).
% subringE(2)
thf(fact_495_subringE_I2_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).
% subringE(2)
thf(fact_496_subringE_I7_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subring_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( add_a_b @ R @ H1 @ H2 ) @ H ) ) ) ) ).
% subringE(7)
thf(fact_497_subringE_I7_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H2 ) @ H ) ) ) ) ).
% subringE(7)
thf(fact_498_subringE_I6_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subring_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H2 ) @ H ) ) ) ) ).
% subringE(6)
thf(fact_499_subringE_I6_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H2 ) @ H ) ) ) ) ).
% subringE(6)
thf(fact_500_subfieldE_I1_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( subring_a_b @ K @ R ) ) ).
% subfieldE(1)
thf(fact_501_subfieldE_I1_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( subrin6918843898125473962t_unit @ K @ R ) ) ).
% subfieldE(1)
thf(fact_502_subringE_I3_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H ) ) ).
% subringE(3)
thf(fact_503_subringE_I3_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ H ) ) ).
% subringE(3)
thf(fact_504_subringE_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subringE(1)
thf(fact_505_subringE_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subringE(1)
thf(fact_506_subringE_I1_J,axiom,
! [H: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subrin3541368690557094692t_unit @ H @ R )
=> ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% subringE(1)
thf(fact_507_ring_Ois__root_Ocong,axiom,
polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).
% ring.is_root.cong
thf(fact_508_ring_Ois__root_Ocong,axiom,
polyno6951661231331188332t_unit = polyno6951661231331188332t_unit ).
% ring.is_root.cong
thf(fact_509_ring_Opdiv_Ocong,axiom,
polynomial_pdiv_a_b = polynomial_pdiv_a_b ).
% ring.pdiv.cong
thf(fact_510_ring_Olong__divides_Ocong,axiom,
polyno2806191415236617128es_a_b = polyno2806191415236617128es_a_b ).
% ring.long_divides.cong
thf(fact_511_pdivides__def,axiom,
( polyno5814909790663948098es_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b] : ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).
% pdivides_def
thf(fact_512_pdivides__def,axiom,
( polyno8016796738000020810t_unit
= ( ^ [R2: partia2670972154091845814t_unit] : ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ).
% pdivides_def
thf(fact_513_pdivides__def,axiom,
( polyno4453881341673752516t_unit
= ( ^ [R2: partia2956882679547061052t_unit] : ( factor652753743487153968t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) ) ) ).
% pdivides_def
thf(fact_514_x_Oexp__base__closed,axiom,
! [X3: list_a,N: nat] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( polyno3522816881121920896t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ N ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.exp_base_closed
thf(fact_515_x_Oeval__monom,axiom,
! [B: list_a,A2: list_a,N: nat] :
( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ N ) @ A2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ N ) ) ) ) ) ).
% x.eval_monom
thf(fact_516_eval__rewrite,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( P
= ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ K ) @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ P ) @ ( var_a_b @ r ) ) ) ) ) ).
% eval_rewrite
thf(fact_517_x_Ofactors__dividesI,axiom,
! [Fs: list_list_a,A2: list_a,F: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fs @ A2 )
=> ( ( member_list_a @ F @ ( set_list_a2 @ Fs ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) ) ) ) ).
% x.factors_dividesI
thf(fact_518_x_Oee__trans,axiom,
! [As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
=> ( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Bs @ Cs )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Cs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Cs ) ) ) ) ) ) ).
% x.ee_trans
thf(fact_519_x_Oee__sym,axiom,
! [As: list_list_a,Bs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Bs @ As ) ) ) ) ).
% x.ee_sym
thf(fact_520_x_Ocarrier__is__subring,axiom,
subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.carrier_is_subring
thf(fact_521_eval__var,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X3 )
= X3 ) ) ).
% eval_var
thf(fact_522_var__closed_I1_J,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( member_list_a @ ( var_a_b @ r ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).
% var_closed(1)
thf(fact_523_var__pow__closed,axiom,
! [K: set_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).
% var_pow_closed
thf(fact_524_x_Ofactors__closed,axiom,
! [Fs: list_list_a,A2: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fs @ A2 )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.factors_closed
thf(fact_525_x_Ocarrier__polynomial__shell,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.carrier_polynomial_shell
thf(fact_526_x_Oee__refl,axiom,
! [As: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ As ) ) ).
% x.ee_refl
thf(fact_527_x_Omonom__in__carrier,axiom,
! [A2: list_a,N: nat] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ N ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.monom_in_carrier
thf(fact_528_ring_Omonom_Ocong,axiom,
monom_7446464087056152608t_unit = monom_7446464087056152608t_unit ).
% ring.monom.cong
thf(fact_529_ring_Omonom_Ocong,axiom,
monom_a_b = monom_a_b ).
% ring.monom.cong
thf(fact_530_unitary__monom__eq__var__pow,axiom,
! [K: set_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( monom_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) ) ) ).
% unitary_monom_eq_var_pow
thf(fact_531_x_Omultlist__dividesI,axiom,
! [F: list_a,Fs: list_list_a] :
( ( member_list_a @ F @ ( set_list_a2 @ Fs ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ ( foldr_list_a_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Fs @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.multlist_dividesI
thf(fact_532_x_OfactorsI,axiom,
! [Fs: list_list_a,A2: list_a] :
( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Fs ) )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
=> ( ( ( foldr_list_a_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Fs @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= A2 )
=> ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fs @ A2 ) ) ) ).
% x.factorsI
thf(fact_533_x_Ofactors__mult__single,axiom,
! [A2: list_a,Fb: list_list_a,B: list_a] :
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 )
=> ( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fb @ B )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ A2 @ Fb ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) ) ) ) ) ).
% x.factors_mult_single
thf(fact_534_x_Ofactors__mult,axiom,
! [Fa: list_list_a,A2: list_a,Fb: list_list_a,B: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fa @ A2 )
=> ( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fb @ B )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fa ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fb ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ Fa @ Fb ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) ) ) ) ) ) ).
% x.factors_mult
thf(fact_535_x_Onormalize_Ocases,axiom,
! [X3: list_list_a] :
( ( X3 != nil_list_a )
=> ~ ! [V2: list_a,Va: list_list_a] :
( X3
!= ( cons_list_a @ V2 @ Va ) ) ) ).
% x.normalize.cases
thf(fact_536_x_Oconst__term__eq__last,axiom,
! [P: list_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ ( cons_list_a @ A2 @ nil_list_a ) ) )
= A2 ) ) ) ).
% x.const_term_eq_last
thf(fact_537_x_Oconst__term__explicit,axiom,
! [P: list_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= A2 )
=> ~ ! [P4: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P4 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( P
!= ( append_list_a @ P4 @ ( cons_list_a @ A2 @ nil_list_a ) ) ) ) ) ) ) ).
% x.const_term_explicit
thf(fact_538_x_Oeval__append__aux,axiom,
! [P: list_list_a,B: list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ ( cons_list_a @ B @ nil_list_a ) ) @ A2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A2 ) @ A2 ) @ B ) ) ) ) ) ).
% x.eval_append_aux
thf(fact_539_x_Oadd_Omultlist__closed,axiom,
! [Fs: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( foldr_list_a_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Fs @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.multlist_closed
thf(fact_540_x_Omultlist__closed,axiom,
! [Fs: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( foldr_list_a_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Fs @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.multlist_closed
thf(fact_541_univ__poly__one,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ K ) )
= ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) ) ).
% univ_poly_one
thf(fact_542_univ__poly__one,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) )
= ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ nil_list_a ) ) ).
% univ_poly_one
thf(fact_543_var__def,axiom,
( var_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b] : ( cons_a @ ( one_a_ring_ext_a_b @ R2 ) @ ( cons_a @ ( zero_a_b @ R2 ) @ nil_a ) ) ) ) ).
% var_def
thf(fact_544_var__def,axiom,
( var_li8453953174693405341t_unit
= ( ^ [R2: partia2670972154091845814t_unit] : ( cons_list_a @ ( one_li8328186300101108157t_unit @ R2 ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ nil_list_a ) ) ) ) ).
% var_def
thf(fact_545_factors__def,axiom,
( factor5638265376665762323xt_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b,Fs2: list_a,A4: a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Fs2 ) )
=> ( irredu6211895646901577903xt_a_b @ G2 @ X2 ) )
& ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ G2 ) @ Fs2 @ ( one_a_ring_ext_a_b @ G2 ) )
= A4 ) ) ) ) ).
% factors_def
thf(fact_546_factors__def,axiom,
( factor7181967632740204193t_unit
= ( ^ [G2: partia2670972154091845814t_unit,Fs2: list_list_a,A4: list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Fs2 ) )
=> ( irredu4230924414530676029t_unit @ G2 @ X2 ) )
& ( ( foldr_list_a_list_a @ ( mult_l7073676228092353617t_unit @ G2 ) @ Fs2 @ ( one_li8328186300101108157t_unit @ G2 ) )
= A4 ) ) ) ) ).
% factors_def
thf(fact_547_factors__def,axiom,
( factor4979495158039764464t_unit
= ( ^ [G2: partia8223610829204095565t_unit,Fs2: list_a,A4: a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Fs2 ) )
=> ( irredu4023057619401689684t_unit @ G2 @ X2 ) )
& ( ( foldr_a_a @ ( mult_a_Product_unit @ G2 ) @ Fs2 @ ( one_a_Product_unit @ G2 ) )
= A4 ) ) ) ) ).
% factors_def
thf(fact_548_factorsE,axiom,
! [G: partia2175431115845679010xt_a_b,Fs: list_a,A2: a] :
( ( factor5638265376665762323xt_a_b @ G @ Fs @ A2 )
=> ~ ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Fs ) )
=> ( irredu6211895646901577903xt_a_b @ G @ X4 ) )
=> ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ G ) @ Fs @ ( one_a_ring_ext_a_b @ G ) )
!= A2 ) ) ) ).
% factorsE
thf(fact_549_factorsE,axiom,
! [G: partia2670972154091845814t_unit,Fs: list_list_a,A2: list_a] :
( ( factor7181967632740204193t_unit @ G @ Fs @ A2 )
=> ~ ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Fs ) )
=> ( irredu4230924414530676029t_unit @ G @ X4 ) )
=> ( ( foldr_list_a_list_a @ ( mult_l7073676228092353617t_unit @ G ) @ Fs @ ( one_li8328186300101108157t_unit @ G ) )
!= A2 ) ) ) ).
% factorsE
thf(fact_550_factorsE,axiom,
! [G: partia8223610829204095565t_unit,Fs: list_a,A2: a] :
( ( factor4979495158039764464t_unit @ G @ Fs @ A2 )
=> ~ ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Fs ) )
=> ( irredu4023057619401689684t_unit @ G @ X4 ) )
=> ( ( foldr_a_a @ ( mult_a_Product_unit @ G ) @ Fs @ ( one_a_Product_unit @ G ) )
!= A2 ) ) ) ).
% factorsE
thf(fact_551_x_Oeval__append,axiom,
! [P: list_list_a,Q: list_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ Q ) @ A2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A2 ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ ( size_s349497388124573686list_a @ Q ) ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ A2 ) ) ) ) ) ) ).
% x.eval_append
thf(fact_552_append1__eq__conv,axiom,
! [Xs: list_list_a,X3: list_a,Ys: list_list_a,Y: list_a] :
( ( ( append_list_a @ Xs @ ( cons_list_a @ X3 @ nil_list_a ) )
= ( append_list_a @ Ys @ ( cons_list_a @ Y @ nil_list_a ) ) )
= ( ( Xs = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_553_append1__eq__conv,axiom,
! [Xs: list_a,X3: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X3 @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_554_x_Omultlist__perm__cong,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_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( foldr_list_a_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ As @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( foldr_list_a_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Bs @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.multlist_perm_cong
thf(fact_555_x_Oadd_Omultlist__perm__cong,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_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( foldr_list_a_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ As @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( foldr_list_a_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Bs @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.multlist_perm_cong
thf(fact_556_x_Omonic__degree__one__root__condition,axiom,
! [A2: list_a,B: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 ) @ nil_list_a ) ) @ B )
= ( A2 = B ) ) ) ).
% x.monic_degree_one_root_condition
thf(fact_557_normalize_Ocases,axiom,
! [X3: list_a] :
( ( X3 != nil_a )
=> ~ ! [V2: a,Va: list_a] :
( X3
!= ( cons_a @ V2 @ Va ) ) ) ).
% normalize.cases
thf(fact_558_combine_Ocases,axiom,
! [X3: produc9164743771328383783list_a] :
( ! [K4: a,Ks: list_a,U2: a,Us: list_a] :
( X3
!= ( produc6837034575241423639list_a @ ( cons_a @ K4 @ Ks ) @ ( cons_a @ U2 @ Us ) ) )
=> ( ! [Us: list_a] :
( X3
!= ( produc6837034575241423639list_a @ nil_a @ Us ) )
=> ~ ! [Ks: list_a] :
( X3
!= ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).
% combine.cases
thf(fact_559_poly__mult_Ocases,axiom,
! [X3: produc9164743771328383783list_a] :
( ! [P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ nil_a @ P22 ) )
=> ~ ! [V2: a,Va: list_a,P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ ( cons_a @ V2 @ Va ) @ P22 ) ) ) ).
% poly_mult.cases
thf(fact_560_factors__closed,axiom,
! [Fs: list_a,A2: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fs @ A2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% factors_closed
thf(fact_561_eval__in__carrier,axiom,
! [P: list_a,X3: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier
thf(fact_562_factors__mult,axiom,
! [Fa: list_a,A2: a,Fb: list_a,B: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fa @ A2 )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) ) ) ) ) ) ).
% factors_mult
thf(fact_563_factors__dividesI,axiom,
! [Fs: list_a,A2: a,F: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fs @ A2 )
=> ( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ F @ A2 ) ) ) ) ).
% factors_dividesI
thf(fact_564_const__term__simprules_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( const_term_a_b @ r @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% const_term_simprules(1)
thf(fact_565_factors__mult__single,axiom,
! [A2: a,Fb: list_a,B: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A2 )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( cons_a @ A2 @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) ) ) ) ) ).
% factors_mult_single
thf(fact_566_factorsI,axiom,
! [Fs: list_a,A2: a] :
( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Fs ) )
=> ( irredu6211895646901577903xt_a_b @ r @ X ) )
=> ( ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) )
= A2 )
=> ( factor5638265376665762323xt_a_b @ r @ Fs @ A2 ) ) ) ).
% factorsI
thf(fact_567_x_Osubring__props_I5_J,axiom,
! [K: set_list_a,H3: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H3 @ K )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H3 ) @ K ) ) ) ).
% x.subring_props(5)
thf(fact_568_const__term__eq__last,axiom,
! [P: list_a,A2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( append_a @ P @ ( cons_a @ A2 @ nil_a ) ) )
= A2 ) ) ) ).
% const_term_eq_last
thf(fact_569_const__term__explicit,axiom,
! [P: list_a,A2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= A2 )
=> ~ ! [P4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( P
!= ( append_a @ P4 @ ( cons_a @ A2 @ nil_a ) ) ) ) ) ) ) ).
% const_term_explicit
thf(fact_570_x_Oee__length,axiom,
! [As: list_list_a,Bs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
=> ( ( size_s349497388124573686list_a @ As )
= ( size_s349497388124573686list_a @ Bs ) ) ) ).
% x.ee_length
thf(fact_571_x_Oa__transpose__inv,axiom,
! [X3: list_a,Y: list_a,Z: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y )
= Z )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) @ Z )
= Y ) ) ) ) ) ).
% x.a_transpose_inv
thf(fact_572_x_Oadd_Oinv__mult__group,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) ) ) ) ) ).
% x.add.inv_mult_group
thf(fact_573_x_Oadd_Oinv__solve__left,axiom,
! [A2: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B ) @ C ) )
= ( C
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A2 ) ) ) ) ) ) ).
% x.add.inv_solve_left
thf(fact_574_x_Oadd_Oinv__solve__left_H,axiom,
! [A2: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B ) @ C )
= A2 )
= ( C
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A2 ) ) ) ) ) ) ).
% x.add.inv_solve_left'
thf(fact_575_x_Oadd_Oinv__solve__right,axiom,
! [A2: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C ) ) )
= ( B
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ C ) ) ) ) ) ) ).
% x.add.inv_solve_right
thf(fact_576_x_Oadd_Oinv__solve__right_H,axiom,
! [A2: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C ) )
= A2 )
= ( B
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ C ) ) ) ) ) ) ).
% x.add.inv_solve_right'
thf(fact_577_x_Ominus__add,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) ) ) ) ) ).
% x.minus_add
thf(fact_578_x_Or__neg1,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) )
= Y ) ) ) ).
% x.r_neg1
thf(fact_579_x_Or__neg2,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) @ Y ) )
= Y ) ) ) ).
% x.r_neg2
thf(fact_580_x_Ol__minus,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) @ Y )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) ) ) ) ) ).
% x.l_minus
thf(fact_581_x_Or__minus,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) ) ) ) ) ).
% x.r_minus
thf(fact_582_univ__poly__a__inv__consistent,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ) ).
% univ_poly_a_inv_consistent
thf(fact_583_x_Oadd_Oint__pow__inv,axiom,
! [X3: list_a,I2: int] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ X3 ) ) ) ) ).
% x.add.int_pow_inv
thf(fact_584_multlist__dividesI,axiom,
! [F: a,Fs: list_a] :
( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ F @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% multlist_dividesI
thf(fact_585_x_Ominus__eq,axiom,
! [X3: list_a,Y: list_a] :
( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) ) ) ).
% x.minus_eq
thf(fact_586_long__division__a__inv_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% long_division_a_inv(2)
thf(fact_587_long__division__a__inv_I1_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% long_division_a_inv(1)
thf(fact_588_x_Oirrlist__perm__cong,axiom,
! [As: list_list_a,Bs: list_list_a] :
( ( ( mset_list_a @ As )
= ( mset_list_a @ Bs ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ As ) )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Bs ) )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X4 ) ) ) ) ).
% x.irrlist_perm_cong
thf(fact_589_x_Ol__neg,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.l_neg
thf(fact_590_x_Ominus__equality,axiom,
! [Y: list_a,X3: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 )
= Y ) ) ) ) ).
% x.minus_equality
thf(fact_591_x_Or__neg,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.r_neg
thf(fact_592_x_Osum__zero__eq__neg,axiom,
! [X3: list_a,Y: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( X3
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) ) ) ) ) ).
% x.sum_zero_eq_neg
thf(fact_593_eval__append__aux,axiom,
! [P: list_a,B: a,A2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P @ ( cons_a @ B @ nil_a ) ) @ A2 )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A2 ) @ A2 ) @ B ) ) ) ) ) ).
% eval_append_aux
thf(fact_594_x_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_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.perm_closed
thf(fact_595_s_Oring_Oa__inv__closed,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_a @ ( eval_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) @ s2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% s.ring.a_inv_closed
thf(fact_596_List_Ofinite__set,axiom,
! [Xs: list_a] : ( finite_finite_a @ ( set_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_597_List_Ofinite__set,axiom,
! [Xs: list_list_a] : ( finite_finite_list_a @ ( set_list_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_598_append_Oright__neutral,axiom,
! [A2: list_a] :
( ( append_a @ A2 @ nil_a )
= A2 ) ).
% append.right_neutral
thf(fact_599_append_Oright__neutral,axiom,
! [A2: list_list_a] :
( ( append_list_a @ A2 @ nil_list_a )
= A2 ) ).
% append.right_neutral
thf(fact_600_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_601_append__Nil2,axiom,
! [Xs: list_list_a] :
( ( append_list_a @ Xs @ nil_list_a )
= Xs ) ).
% append_Nil2
thf(fact_602_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_603_append__self__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_a ) ) ).
% append_self_conv
thf(fact_604_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_605_self__append__conv,axiom,
! [Y: list_list_a,Ys: list_list_a] :
( ( Y
= ( append_list_a @ Y @ Ys ) )
= ( Ys = nil_list_a ) ) ).
% self_append_conv
thf(fact_606_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_607_append__self__conv2,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_a ) ) ).
% append_self_conv2
thf(fact_608_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_609_self__append__conv2,axiom,
! [Y: list_list_a,Xs: list_list_a] :
( ( Y
= ( append_list_a @ Xs @ Y ) )
= ( Xs = nil_list_a ) ) ).
% self_append_conv2
thf(fact_610_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_611_Nil__is__append__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( nil_list_a
= ( append_list_a @ Xs @ Ys ) )
= ( ( Xs = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% Nil_is_append_conv
thf(fact_612_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_613_append__is__Nil__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( append_list_a @ Xs @ Ys )
= nil_list_a )
= ( ( Xs = nil_list_a )
& ( Ys = nil_list_a ) ) ) ).
% append_is_Nil_conv
thf(fact_614_list_Omap__disc__iff,axiom,
! [F: a > a,A2: list_a] :
( ( ( map_a_a @ F @ A2 )
= nil_a )
= ( A2 = nil_a ) ) ).
% list.map_disc_iff
thf(fact_615_list_Omap__disc__iff,axiom,
! [F: list_a > a,A2: list_list_a] :
( ( ( map_list_a_a @ F @ A2 )
= nil_a )
= ( A2 = nil_list_a ) ) ).
% list.map_disc_iff
thf(fact_616_list_Omap__disc__iff,axiom,
! [F: a > list_a,A2: list_a] :
( ( ( map_a_list_a @ F @ A2 )
= nil_list_a )
= ( A2 = nil_a ) ) ).
% list.map_disc_iff
thf(fact_617_list_Omap__disc__iff,axiom,
! [F: list_a > list_a,A2: list_list_a] :
( ( ( map_list_a_list_a @ F @ A2 )
= nil_list_a )
= ( A2 = nil_list_a ) ) ).
% list.map_disc_iff
thf(fact_618_Nil__is__map__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( nil_a
= ( map_a_a @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_619_Nil__is__map__conv,axiom,
! [F: list_a > a,Xs: list_list_a] :
( ( nil_a
= ( map_list_a_a @ F @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% Nil_is_map_conv
thf(fact_620_Nil__is__map__conv,axiom,
! [F: a > list_a,Xs: list_a] :
( ( nil_list_a
= ( map_a_list_a @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_621_Nil__is__map__conv,axiom,
! [F: list_a > list_a,Xs: list_list_a] :
( ( nil_list_a
= ( map_list_a_list_a @ F @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% Nil_is_map_conv
thf(fact_622_map__is__Nil__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( ( map_a_a @ F @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_623_map__is__Nil__conv,axiom,
! [F: list_a > a,Xs: list_list_a] :
( ( ( map_list_a_a @ F @ Xs )
= nil_a )
= ( Xs = nil_list_a ) ) ).
% map_is_Nil_conv
thf(fact_624_map__is__Nil__conv,axiom,
! [F: a > list_a,Xs: list_a] :
( ( ( map_a_list_a @ F @ Xs )
= nil_list_a )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_625_map__is__Nil__conv,axiom,
! [F: list_a > list_a,Xs: list_list_a] :
( ( ( map_list_a_list_a @ F @ Xs )
= nil_list_a )
= ( Xs = nil_list_a ) ) ).
% map_is_Nil_conv
thf(fact_626_perm__map,axiom,
! [A2: list_list_a,B: list_list_a,F: list_a > list_a] :
( ( ( mset_list_a @ A2 )
= ( mset_list_a @ B ) )
=> ( ( mset_list_a @ ( map_list_a_list_a @ F @ A2 ) )
= ( mset_list_a @ ( map_list_a_list_a @ F @ B ) ) ) ) ).
% perm_map
thf(fact_627_perm__map,axiom,
! [A2: list_list_a,B: list_list_a,F: list_a > a] :
( ( ( mset_list_a @ A2 )
= ( mset_list_a @ B ) )
=> ( ( mset_a @ ( map_list_a_a @ F @ A2 ) )
= ( mset_a @ ( map_list_a_a @ F @ B ) ) ) ) ).
% perm_map
thf(fact_628_perm__map,axiom,
! [A2: list_a,B: list_a,F: a > list_a] :
( ( ( mset_a @ A2 )
= ( mset_a @ B ) )
=> ( ( mset_list_a @ ( map_a_list_a @ F @ A2 ) )
= ( mset_list_a @ ( map_a_list_a @ F @ B ) ) ) ) ).
% perm_map
thf(fact_629_perm__map,axiom,
! [A2: list_a,B: list_a,F: a > a] :
( ( ( mset_a @ A2 )
= ( mset_a @ B ) )
=> ( ( mset_a @ ( map_a_a @ F @ A2 ) )
= ( mset_a @ ( map_a_a @ F @ B ) ) ) ) ).
% perm_map
thf(fact_630_poly__mult__var,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( P = nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ ( var_a_b @ r ) )
= nil_a ) )
& ( ( P != nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ ( var_a_b @ r ) )
= ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ).
% poly_mult_var
thf(fact_631_x_OsubringI,axiom,
! [H: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ H )
=> ( ! [H4: list_a] :
( ( member_list_a @ H4 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H4 ) @ H ) )
=> ( ! [H12: list_a,H22: list_a] :
( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H12 @ H22 ) @ H ) ) )
=> ( ! [H12: list_a,H22: list_a] :
( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H12 @ H22 ) @ H ) ) )
=> ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.subringI
thf(fact_632_monom__in__carrier,axiom,
! [A2: a,N: nat] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( monom_a_b @ r @ A2 @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% monom_in_carrier
thf(fact_633_add_Omultlist__closed,axiom,
! [Fs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( foldr_a_a @ ( add_a_b @ r ) @ Fs @ ( zero_a_b @ r ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.multlist_closed
thf(fact_634_x_Oadd_Oinv__closed,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.inv_closed
thf(fact_635_x_Ominus__minus,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) )
= X3 ) ) ).
% x.minus_minus
thf(fact_636_x_Ominus__zero,axiom,
( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.minus_zero
thf(fact_637_multlist__closed,axiom,
! [Fs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% multlist_closed
thf(fact_638_x_Oadd_Oinv__eq__1__iff,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( X3
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.inv_eq_1_iff
thf(fact_639_x_OUnits__minus__one__closed,axiom,
member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.Units_minus_one_closed
thf(fact_640_subringE_I5_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H3: a] :
( ( subring_a_b @ H @ R )
=> ( ( member_a @ H3 @ H )
=> ( member_a @ ( a_inv_a_b @ R @ H3 ) @ H ) ) ) ).
% subringE(5)
thf(fact_641_subringE_I5_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H3: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ( member_list_a @ H3 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H3 ) @ H ) ) ) ).
% subringE(5)
thf(fact_642_perm__map__switch,axiom,
! [F: list_a > list_a,A2: list_list_a,B: list_list_a,C: list_list_a] :
( ( ( map_list_a_list_a @ F @ A2 )
= ( map_list_a_list_a @ F @ B ) )
=> ( ( ( mset_list_a @ B )
= ( mset_list_a @ C ) )
=> ? [D2: list_list_a] :
( ( ( mset_list_a @ A2 )
= ( mset_list_a @ D2 ) )
& ( ( map_list_a_list_a @ F @ D2 )
= ( map_list_a_list_a @ F @ C ) ) ) ) ) ).
% perm_map_switch
thf(fact_643_perm__map__switch,axiom,
! [F: a > list_a,A2: list_a,B: list_a,C: list_a] :
( ( ( map_a_list_a @ F @ A2 )
= ( map_a_list_a @ F @ B ) )
=> ( ( ( mset_a @ B )
= ( mset_a @ C ) )
=> ? [D2: list_a] :
( ( ( mset_a @ A2 )
= ( mset_a @ D2 ) )
& ( ( map_a_list_a @ F @ D2 )
= ( map_a_list_a @ F @ C ) ) ) ) ) ).
% perm_map_switch
thf(fact_644_perm__setP,axiom,
! [As: list_list_a,Bs: list_list_a,P2: set_list_a > $o] :
( ( ( mset_list_a @ As )
= ( mset_list_a @ Bs ) )
=> ( ( P2 @ ( set_list_a2 @ As ) )
=> ( P2 @ ( set_list_a2 @ Bs ) ) ) ) ).
% perm_setP
thf(fact_645_perm__setP,axiom,
! [As: list_a,Bs: list_a,P2: set_a > $o] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( P2 @ ( set_a2 @ As ) )
=> ( P2 @ ( set_a2 @ Bs ) ) ) ) ).
% perm_setP
thf(fact_646_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P2: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y5: a,Ys2: list_a,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_647_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,Ws: list_a,P2: list_list_a > list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y5: a,Ys2: list_a,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_648_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,Ws: list_a,P2: list_a > list_list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y5: list_a,Ys2: list_list_a,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_649_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,Ws: list_a,P2: list_a > list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y5: a,Ys2: list_a,Z2: list_a,Zs2: list_list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_list_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_650_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_list_a,P2: list_a > list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X: a,Xs2: list_a,Y5: a,Ys2: list_a,Z2: a,Zs2: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_651_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_a,Ws: list_a,P2: list_list_a > list_list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_list_a @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y5: list_a,Ys2: list_list_a,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_652_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_list_a,Ws: list_a,P2: list_list_a > list_a > list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y5: a,Ys2: list_a,Z2: list_a,Zs2: list_list_a,W: a,Ws2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_list_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_653_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,Ws: list_list_a,P2: list_list_a > list_a > list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y5: a,Ys2: list_a,Z2: a,Zs2: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_654_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_list_a,Ws: list_a,P2: list_a > list_list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_list_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y5: list_a,Ys2: list_list_a,Z2: list_a,Zs2: list_list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys2 ) @ ( cons_list_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_655_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,Ws: list_list_a,P2: list_a > list_list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_a @ nil_list_a )
=> ( ! [X: a,Xs2: list_a,Y5: list_a,Ys2: list_list_a,Z2: a,Zs2: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_656_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a,P2: list_list_a > list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_list_a @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y5: list_a,Ys2: list_list_a,Z2: list_a,Zs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys2 ) @ ( cons_list_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_657_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_a,P2: list_list_a > list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_list_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y5: list_a,Ys2: list_list_a,Z2: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_658_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_list_a,P2: list_list_a > list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y5: a,Ys2: list_a,Z2: list_a,Zs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_list_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_659_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,P2: list_list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y5: a,Ys2: list_a,Z2: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_660_list__induct3,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_list_a,P2: list_a > list_list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_list_a )
=> ( ! [X: a,Xs2: list_a,Y5: list_a,Ys2: list_list_a,Z2: list_a,Zs2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys2 ) @ ( cons_list_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_661_list__induct3,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,P2: list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y5: list_a,Ys2: list_list_a,Z2: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_662_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,P2: list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X: a,Xs2: list_a,Y5: a,Ys2: list_a,Z2: list_a,Zs2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_list_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_663_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P2: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y5: a,Ys2: list_a,Z2: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y5 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_664_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_list_a,P2: list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P2 @ nil_list_a @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y5: list_a,Ys2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_665_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_a,P2: list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_list_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a,Y5: a,Ys2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y5 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_666_list__induct2,axiom,
! [Xs: list_a,Ys: list_list_a,P2: list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_list_a )
=> ( ! [X: a,Xs2: list_a,Y5: list_a,Ys2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_667_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P2: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a,Y5: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y5 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_668_a__minus__def,axiom,
( a_minu3984020753470702548t_unit
= ( ^ [R2: partia2670972154091845814t_unit,X2: list_a,Y4: list_a] : ( add_li7652885771158616974t_unit @ R2 @ X2 @ ( a_inv_8944721093294617173t_unit @ R2 @ Y4 ) ) ) ) ).
% a_minus_def
thf(fact_669_a__minus__def,axiom,
( a_minus_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,X2: a,Y4: a] : ( add_a_b @ R2 @ X2 @ ( a_inv_a_b @ R2 @ Y4 ) ) ) ) ).
% a_minus_def
thf(fact_670_same__length__different,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs != Ys )
=> ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ? [Pre: list_list_a,X: list_a,Xs3: list_list_a,Y5: list_a,Ys3: list_list_a] :
( ( X != Y5 )
& ( Xs
= ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ X @ nil_list_a ) @ Xs3 ) ) )
& ( Ys
= ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ Y5 @ nil_list_a ) @ Ys3 ) ) ) ) ) ) ).
% same_length_different
thf(fact_671_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X: a,Xs3: list_a,Y5: a,Ys3: list_a] :
( ( X != Y5 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X @ nil_a ) @ Xs3 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y5 @ nil_a ) @ Ys3 ) ) ) ) ) ) ).
% same_length_different
thf(fact_672_subset__code_I1_J,axiom,
! [Xs: list_list_list_a,B4: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Xs ) @ B4 )
= ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( set_list_list_a2 @ Xs ) )
=> ( member_list_list_a @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_673_subset__code_I1_J,axiom,
! [Xs: list_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B4 )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( member_a @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_674_subset__code_I1_J,axiom,
! [Xs: list_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B4 )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_675_finite__list,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ? [Xs2: list_a] :
( ( set_a2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_676_finite__list,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ? [Xs2: list_list_a] :
( ( set_list_a2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_677_list__nonempty__induct,axiom,
! [Xs: list_list_a,P2: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X: list_a] : ( P2 @ ( cons_list_a @ X @ nil_list_a ) )
=> ( ! [X: list_a,Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_678_list__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X: a] : ( P2 @ ( cons_a @ X @ nil_a ) )
=> ( ! [X: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_679_list__induct2_H,axiom,
! [P2: list_list_a > list_list_a > $o,Xs: list_list_a,Ys: list_list_a] :
( ( P2 @ nil_list_a @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a] : ( P2 @ ( cons_list_a @ X @ Xs2 ) @ nil_list_a )
=> ( ! [Y5: list_a,Ys2: list_list_a] : ( P2 @ nil_list_a @ ( cons_list_a @ Y5 @ Ys2 ) )
=> ( ! [X: list_a,Xs2: list_list_a,Y5: list_a,Ys2: list_list_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_680_list__induct2_H,axiom,
! [P2: list_list_a > list_a > $o,Xs: list_list_a,Ys: list_a] :
( ( P2 @ nil_list_a @ nil_a )
=> ( ! [X: list_a,Xs2: list_list_a] : ( P2 @ ( cons_list_a @ X @ Xs2 ) @ nil_a )
=> ( ! [Y5: a,Ys2: list_a] : ( P2 @ nil_list_a @ ( cons_a @ Y5 @ Ys2 ) )
=> ( ! [X: list_a,Xs2: list_list_a,Y5: a,Ys2: list_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_list_a @ X @ Xs2 ) @ ( cons_a @ Y5 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_681_list__induct2_H,axiom,
! [P2: list_a > list_list_a > $o,Xs: list_a,Ys: list_list_a] :
( ( P2 @ nil_a @ nil_list_a )
=> ( ! [X: a,Xs2: list_a] : ( P2 @ ( cons_a @ X @ Xs2 ) @ nil_list_a )
=> ( ! [Y5: list_a,Ys2: list_list_a] : ( P2 @ nil_a @ ( cons_list_a @ Y5 @ Ys2 ) )
=> ( ! [X: a,Xs2: list_a,Y5: list_a,Ys2: list_list_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_682_list__induct2_H,axiom,
! [P2: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P2 @ nil_a @ nil_a )
=> ( ! [X: a,Xs2: list_a] : ( P2 @ ( cons_a @ X @ Xs2 ) @ nil_a )
=> ( ! [Y5: a,Ys2: list_a] : ( P2 @ nil_a @ ( cons_a @ Y5 @ Ys2 ) )
=> ( ! [X: a,Xs2: list_a,Y5: a,Ys2: list_a] :
( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y5 @ Ys2 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_683_neq__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
= ( ? [Y4: list_a,Ys4: list_list_a] :
( Xs
= ( cons_list_a @ Y4 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_684_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y4: a,Ys4: list_a] :
( Xs
= ( cons_a @ Y4 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_685_remdups__adj_Ocases,axiom,
! [X3: list_list_a] :
( ( X3 != nil_list_a )
=> ( ! [X: list_a] :
( X3
!= ( cons_list_a @ X @ nil_list_a ) )
=> ~ ! [X: list_a,Y5: list_a,Xs2: list_list_a] :
( X3
!= ( cons_list_a @ X @ ( cons_list_a @ Y5 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_686_remdups__adj_Ocases,axiom,
! [X3: list_a] :
( ( X3 != nil_a )
=> ( ! [X: a] :
( X3
!= ( cons_a @ X @ nil_a ) )
=> ~ ! [X: a,Y5: a,Xs2: list_a] :
( X3
!= ( cons_a @ X @ ( cons_a @ Y5 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_687_list_Oexhaust,axiom,
! [Y: list_list_a] :
( ( Y != nil_list_a )
=> ~ ! [X21: list_a,X22: list_list_a] :
( Y
!= ( cons_list_a @ X21 @ X22 ) ) ) ).
% list.exhaust
thf(fact_688_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X21: a,X22: list_a] :
( Y
!= ( cons_a @ X21 @ X22 ) ) ) ).
% list.exhaust
thf(fact_689_list_OdiscI,axiom,
! [List: list_list_a,X212: list_a,X222: list_list_a] :
( ( List
= ( cons_list_a @ X212 @ X222 ) )
=> ( List != nil_list_a ) ) ).
% list.discI
thf(fact_690_list_OdiscI,axiom,
! [List: list_a,X212: a,X222: list_a] :
( ( List
= ( cons_a @ X212 @ X222 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_691_list_Odistinct_I1_J,axiom,
! [X212: list_a,X222: list_list_a] :
( nil_list_a
!= ( cons_list_a @ X212 @ X222 ) ) ).
% list.distinct(1)
thf(fact_692_list_Odistinct_I1_J,axiom,
! [X212: a,X222: list_a] :
( nil_a
!= ( cons_a @ X212 @ X222 ) ) ).
% list.distinct(1)
thf(fact_693_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_694_eq__Nil__appendI,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_a @ nil_list_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_695_append_Oleft__neutral,axiom,
! [A2: list_a] :
( ( append_a @ nil_a @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_696_append_Oleft__neutral,axiom,
! [A2: list_list_a] :
( ( append_list_a @ nil_list_a @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_697_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_698_append__Nil,axiom,
! [Ys: list_list_a] :
( ( append_list_a @ nil_list_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_699_list_Osimps_I8_J,axiom,
! [F: a > a] :
( ( map_a_a @ F @ nil_a )
= nil_a ) ).
% list.simps(8)
thf(fact_700_list_Osimps_I8_J,axiom,
! [F: list_a > a] :
( ( map_list_a_a @ F @ nil_list_a )
= nil_a ) ).
% list.simps(8)
thf(fact_701_list_Osimps_I8_J,axiom,
! [F: a > list_a] :
( ( map_a_list_a @ F @ nil_a )
= nil_list_a ) ).
% list.simps(8)
thf(fact_702_list_Osimps_I8_J,axiom,
! [F: list_a > list_a] :
( ( map_list_a_list_a @ F @ nil_list_a )
= nil_list_a ) ).
% list.simps(8)
thf(fact_703_set__subset__Cons,axiom,
! [Xs: list_a,X3: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_704_set__subset__Cons,axiom,
! [Xs: list_list_a,X3: list_a] : ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ ( cons_list_a @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_705_rev__nonempty__induct,axiom,
! [Xs: list_list_a,P2: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X: list_a] : ( P2 @ ( cons_list_a @ X @ nil_list_a ) )
=> ( ! [X: list_a,Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append_list_a @ Xs2 @ ( cons_list_a @ X @ nil_list_a ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_706_rev__nonempty__induct,axiom,
! [Xs: list_a,P2: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X: a] : ( P2 @ ( cons_a @ X @ nil_a ) )
=> ( ! [X: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_707_append__eq__Cons__conv,axiom,
! [Ys: list_list_a,Zs: list_list_a,X3: list_a,Xs: list_list_a] :
( ( ( append_list_a @ Ys @ Zs )
= ( cons_list_a @ X3 @ Xs ) )
= ( ( ( Ys = nil_list_a )
& ( Zs
= ( cons_list_a @ X3 @ Xs ) ) )
| ? [Ys5: list_list_a] :
( ( Ys
= ( cons_list_a @ X3 @ Ys5 ) )
& ( ( append_list_a @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_708_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X3: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X3 @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X3 @ Xs ) ) )
| ? [Ys5: list_a] :
( ( Ys
= ( cons_a @ X3 @ Ys5 ) )
& ( ( append_a @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_709_Cons__eq__append__conv,axiom,
! [X3: list_a,Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( cons_list_a @ X3 @ Xs )
= ( append_list_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_a )
& ( ( cons_list_a @ X3 @ Xs )
= Zs ) )
| ? [Ys5: list_list_a] :
( ( ( cons_list_a @ X3 @ Ys5 )
= Ys )
& ( Xs
= ( append_list_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_710_Cons__eq__append__conv,axiom,
! [X3: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X3 @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X3 @ Xs )
= Zs ) )
| ? [Ys5: list_a] :
( ( ( cons_a @ X3 @ Ys5 )
= Ys )
& ( Xs
= ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_711_rev__exhaust,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ~ ! [Ys2: list_list_a,Y5: list_a] :
( Xs
!= ( append_list_a @ Ys2 @ ( cons_list_a @ Y5 @ nil_list_a ) ) ) ) ).
% rev_exhaust
thf(fact_712_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y5: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y5 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_713_rev__induct,axiom,
! [P2: list_list_a > $o,Xs: list_list_a] :
( ( P2 @ nil_list_a )
=> ( ! [X: list_a,Xs2: list_list_a] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append_list_a @ Xs2 @ ( cons_list_a @ X @ nil_list_a ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_714_rev__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ( P2 @ nil_a )
=> ( ! [X: a,Xs2: list_a] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_715_shuffles_Ocases,axiom,
! [X3: produc9164743771328383783list_a] :
( ! [Ys2: list_a] :
( X3
!= ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ( ! [Xs2: list_a] :
( X3
!= ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ~ ! [X: a,Xs2: list_a,Y5: a,Ys2: list_a] :
( X3
!= ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs2 ) @ ( cons_a @ Y5 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_716_shuffles_Ocases,axiom,
! [X3: produc7709606177366032167list_a] :
( ! [Ys2: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ nil_list_a @ Ys2 ) )
=> ( ! [Xs2: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ Xs2 @ nil_list_a ) )
=> ~ ! [X: list_a,Xs2: list_list_a,Y5: list_a,Ys2: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ Y5 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_717_transpose_Ocases,axiom,
! [X3: list_list_list_a] :
( ( X3 != nil_list_list_a )
=> ( ! [Xss: list_list_list_a] :
( X3
!= ( cons_list_list_a @ nil_list_a @ Xss ) )
=> ~ ! [X: list_a,Xs2: list_list_a,Xss: list_list_list_a] :
( X3
!= ( cons_list_list_a @ ( cons_list_a @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_718_transpose_Ocases,axiom,
! [X3: list_list_a] :
( ( X3 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X3
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X: a,Xs2: list_a,Xss: list_list_a] :
( X3
!= ( cons_list_a @ ( cons_a @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_719_exp__base__closed,axiom,
! [X3: a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X3 @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% exp_base_closed
thf(fact_720_x_Oadd_Oone__in__subset,axiom,
! [H: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( H != bot_bot_set_list_a )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ H ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ H )
=> ! [Xa2: list_a] :
( ( member_list_a @ Xa2 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Xa2 ) @ H ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ H ) ) ) ) ) ).
% x.add.one_in_subset
thf(fact_721_perm__sing__eq2,axiom,
! [Y: list_a,Ys: list_list_a] :
( ( ( mset_list_a @ ( cons_list_a @ Y @ nil_list_a ) )
= ( mset_list_a @ Ys ) )
= ( Ys
= ( cons_list_a @ Y @ nil_list_a ) ) ) ).
% perm_sing_eq2
thf(fact_722_perm__sing__eq2,axiom,
! [Y: a,Ys: list_a] :
( ( ( mset_a @ ( cons_a @ Y @ nil_a ) )
= ( mset_a @ Ys ) )
= ( Ys
= ( cons_a @ Y @ nil_a ) ) ) ).
% perm_sing_eq2
thf(fact_723_perm__sing__eq,axiom,
! [Ys: list_list_a,Y: list_a] :
( ( ( mset_list_a @ Ys )
= ( mset_list_a @ ( cons_list_a @ Y @ nil_list_a ) ) )
= ( Ys
= ( cons_list_a @ Y @ nil_list_a ) ) ) ).
% perm_sing_eq
thf(fact_724_perm__sing__eq,axiom,
! [Ys: list_a,Y: a] :
( ( ( mset_a @ Ys )
= ( mset_a @ ( cons_a @ Y @ nil_a ) ) )
= ( Ys
= ( cons_a @ Y @ nil_a ) ) ) ).
% perm_sing_eq
thf(fact_725_x_Ocombine__append__zero,axiom,
! [Us2: list_list_a,Ks2: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ Ks2 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) @ Us2 )
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ Us2 ) ) ) ).
% x.combine_append_zero
thf(fact_726_x_Ocombine_Ocases,axiom,
! [X3: produc7709606177366032167list_a] :
( ! [K4: list_a,Ks: list_list_a,U2: list_a,Us: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ ( cons_list_a @ K4 @ Ks ) @ ( cons_list_a @ U2 @ Us ) ) )
=> ( ! [Us: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ nil_list_a @ Us ) )
=> ~ ! [Ks: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ Ks @ nil_list_a ) ) ) ) ).
% x.combine.cases
thf(fact_727_x_Opoly__mult_Ocases,axiom,
! [X3: produc7709606177366032167list_a] :
( ! [P22: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ nil_list_a @ P22 ) )
=> ~ ! [V2: list_a,Va: list_list_a,P22: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ ( cons_list_a @ V2 @ Va ) @ P22 ) ) ) ).
% x.poly_mult.cases
thf(fact_728_subring__props_I5_J,axiom,
! [K: set_a,H3: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H3 @ K )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ K ) ) ) ).
% subring_props(5)
thf(fact_729_add_Oinv__mult__group,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X3 @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X3 ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_730_add_Oinv__solve__left,axiom,
! [A2: a,B: a,C: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A2
= ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
= ( C
= ( add_a_b @ r @ B @ A2 ) ) ) ) ) ) ).
% add.inv_solve_left
thf(fact_731_add_Oinv__solve__left_H,axiom,
! [A2: a,B: a,C: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
= A2 )
= ( C
= ( add_a_b @ r @ B @ A2 ) ) ) ) ) ) ).
% add.inv_solve_left'
thf(fact_732_add_Oinv__solve__right,axiom,
! [A2: a,B: a,C: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A2
= ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
= ( B
= ( add_a_b @ r @ A2 @ C ) ) ) ) ) ) ).
% add.inv_solve_right
thf(fact_733_add_Oinv__solve__right_H,axiom,
! [A2: a,B: a,C: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
= A2 )
= ( B
= ( add_a_b @ r @ A2 @ C ) ) ) ) ) ) ).
% add.inv_solve_right'
thf(fact_734_a__transpose__inv,axiom,
! [X3: a,Y: a,Z: a] :
( ( ( add_a_b @ r @ X3 @ Y )
= Z )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ Z )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_735_local_Ominus__add,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X3 @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_736_r__neg1,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ ( add_a_b @ r @ X3 @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_737_r__neg2,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_738_l__minus,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ Y )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y ) ) ) ) ) ).
% l_minus
thf(fact_739_r__minus,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( a_inv_a_b @ r @ Y ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y ) ) ) ) ) ).
% r_minus
thf(fact_740_minus__eq,axiom,
! [X3: a,Y: a] :
( ( a_minus_a_b @ r @ X3 @ Y )
= ( add_a_b @ r @ X3 @ ( a_inv_a_b @ r @ Y ) ) ) ).
% minus_eq
thf(fact_741_irrlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ As ) )
=> ( irredu6211895646901577903xt_a_b @ r @ X ) )
=> ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Bs ) )
=> ( irredu6211895646901577903xt_a_b @ r @ X4 ) ) ) ) ).
% irrlist_perm_cong
thf(fact_742_l__neg,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ X3 )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_743_minus__equality,axiom,
! [Y: a,X3: a] :
( ( ( add_a_b @ r @ Y @ X3 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X3 )
= Y ) ) ) ) ).
% minus_equality
thf(fact_744_r__neg,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( a_inv_a_b @ r @ X3 ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_745_sum__zero__eq__neg,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X3 @ Y )
= ( zero_a_b @ r ) )
=> ( X3
= ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% sum_zero_eq_neg
thf(fact_746_perm__closed,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% perm_closed
thf(fact_747_square__eq__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X3 @ X3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( X3
= ( one_a_ring_ext_a_b @ r ) )
| ( X3
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% square_eq_one
thf(fact_748_x_Ocarrier__not__empty,axiom,
( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= bot_bot_set_list_a ) ).
% x.carrier_not_empty
thf(fact_749_x_Osubring__props_I4_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( K != bot_bot_set_list_a ) ) ).
% x.subring_props(4)
thf(fact_750_univ__poly__a__inv__length,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
= ( size_size_list_a @ P ) ) ) ) ).
% univ_poly_a_inv_length
thf(fact_751_x_Ocombine_Osimps_I2_J,axiom,
! [Us2: list_list_a] :
( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ Us2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.combine.simps(2)
thf(fact_752_x_Ocombine_Osimps_I3_J,axiom,
! [Ks2: list_list_a] :
( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ nil_list_a )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.combine.simps(3)
thf(fact_753_const__term__simprules__shell_I4_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
= ( a_inv_a_b @ r @ ( const_term_a_b @ r @ P ) ) ) ) ) ).
% const_term_simprules_shell(4)
thf(fact_754_monic__degree__one__root__condition,axiom,
! [A2: a,B: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A2 ) @ nil_a ) ) @ B )
= ( A2 = B ) ) ) ).
% monic_degree_one_root_condition
thf(fact_755_subringI,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ H )
=> ( ! [H4: a] :
( ( member_a @ H4 @ H )
=> ( member_a @ ( a_inv_a_b @ r @ H4 ) @ H ) )
=> ( ! [H12: a,H22: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H22 ) @ H ) ) )
=> ( ! [H12: a,H22: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( add_a_b @ r @ H12 @ H22 ) @ H ) ) )
=> ( subring_a_b @ H @ r ) ) ) ) ) ) ).
% subringI
thf(fact_756_x_Ocombine_Osimps_I1_J,axiom,
! [K3: list_a,Ks2: list_list_a,U: list_a,Us2: list_list_a] :
( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ K3 @ Ks2 ) @ ( cons_list_a @ U @ Us2 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ U ) @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ Us2 ) ) ) ).
% x.combine.simps(1)
thf(fact_757_add_Omultlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( foldr_a_a @ ( add_a_b @ r ) @ As @ ( zero_a_b @ r ) )
= ( foldr_a_a @ ( add_a_b @ r ) @ Bs @ ( zero_a_b @ r ) ) ) ) ) ).
% add.multlist_perm_cong
thf(fact_758_multlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ As @ ( one_a_ring_ext_a_b @ r ) )
= ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Bs @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% multlist_perm_cong
thf(fact_759_pdivides__imp__is__root,axiom,
! [P: list_a,X3: a] :
( ( P != nil_a )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X3 ) @ nil_a ) ) @ P )
=> ( polyno4133073214067823460ot_a_b @ r @ P @ X3 ) ) ) ) ).
% pdivides_imp_is_root
thf(fact_760_x_Ocombine__eq__eval,axiom,
! [Ks2: list_list_a,X3: list_a] :
( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ ( polyno3522816881121920896t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ ( size_s349497388124573686list_a @ Ks2 ) ) )
= ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ X3 ) ) ).
% x.combine_eq_eval
thf(fact_761_perm__empty2,axiom,
! [Xs: list_list_a] :
( ( ( mset_list_a @ Xs )
= ( mset_list_a @ nil_list_a ) )
= ( Xs = nil_list_a ) ) ).
% perm_empty2
thf(fact_762_perm__empty2,axiom,
! [Xs: list_a] :
( ( ( mset_a @ Xs )
= ( mset_a @ nil_a ) )
= ( Xs = nil_a ) ) ).
% perm_empty2
thf(fact_763_perm__empty,axiom,
! [Xs: list_list_a] :
( ( ( mset_list_a @ nil_list_a )
= ( mset_list_a @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% perm_empty
thf(fact_764_perm__empty,axiom,
! [Xs: list_a] :
( ( ( mset_a @ nil_a )
= ( mset_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% perm_empty
thf(fact_765_x_Ocombine__r__distr,axiom,
! [Ks2: list_list_a,Us2: list_list_a,K3: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ Us2 ) )
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( map_list_a_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 ) @ Ks2 ) @ Us2 ) ) ) ) ) ).
% x.combine_r_distr
thf(fact_766_a__inv__closed,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_767_local_Ominus__minus,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X3 ) )
= X3 ) ) ).
% local.minus_minus
thf(fact_768_x_Ocombine_Oelims,axiom,
! [X3: list_list_a,Xa3: list_list_a,Y: list_a] :
( ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Xa3 )
= Y )
=> ( ! [K4: list_a,Ks: list_list_a] :
( ( X3
= ( cons_list_a @ K4 @ Ks ) )
=> ! [U2: list_a,Us: list_list_a] :
( ( Xa3
= ( cons_list_a @ U2 @ Us ) )
=> ( Y
!= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ U2 ) @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ Us ) ) ) ) )
=> ( ( ( X3 = nil_list_a )
=> ( Y
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ~ ( ( Xa3 = nil_list_a )
=> ( Y
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.combine.elims
thf(fact_769_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_770_is__root__imp__pdivides,axiom,
! [P: list_a,X3: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X3 )
=> ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X3 ) @ nil_a ) ) @ P ) ) ) ).
% is_root_imp_pdivides
thf(fact_771_x_Ocombine__append,axiom,
! [Ks2: list_list_a,Us2: list_list_a,Ks3: list_list_a,Vs: list_list_a] :
( ( ( size_s349497388124573686list_a @ Ks2 )
= ( size_s349497388124573686list_a @ Us2 ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ Us2 ) @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks3 @ Vs ) )
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ Ks2 @ Ks3 ) @ ( append_list_a @ Us2 @ Vs ) ) ) ) ) ) ) ) ).
% x.combine_append
thf(fact_772_set__empty,axiom,
! [Xs: list_list_a] :
( ( ( set_list_a2 @ Xs )
= bot_bot_set_list_a )
= ( Xs = nil_list_a ) ) ).
% set_empty
thf(fact_773_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_774_set__empty2,axiom,
! [Xs: list_list_a] :
( ( bot_bot_set_list_a
= ( set_list_a2 @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% set_empty2
thf(fact_775_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_776_add_Oinv__eq__1__iff,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X3 )
= ( zero_a_b @ r ) )
= ( X3
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_777_Units__minus__one__closed,axiom,
member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_minus_one_closed
thf(fact_778_finsum__empty,axiom,
! [F: list_a > a] :
( ( finsum_a_b_list_a @ r @ F @ bot_bot_set_list_a )
= ( zero_a_b @ r ) ) ).
% finsum_empty
thf(fact_779_finsum__empty,axiom,
! [F: a > a] :
( ( finsum_a_b_a @ r @ F @ bot_bot_set_a )
= ( zero_a_b @ r ) ) ).
% finsum_empty
thf(fact_780_finprod__empty,axiom,
! [F: list_a > a] :
( ( finpro6052973074229812797list_a @ r @ F @ bot_bot_set_list_a )
= ( one_a_ring_ext_a_b @ r ) ) ).
% finprod_empty
thf(fact_781_finprod__empty,axiom,
! [F: a > a] :
( ( finpro205304725090349623_a_b_a @ r @ F @ bot_bot_set_a )
= ( one_a_ring_ext_a_b @ r ) ) ).
% finprod_empty
thf(fact_782_x_Ofinsum__empty,axiom,
! [F: list_a > list_a] :
( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ bot_bot_set_list_a )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.finsum_empty
thf(fact_783_x_Ofinsum__empty,axiom,
! [F: a > list_a] :
( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ bot_bot_set_a )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.finsum_empty
thf(fact_784_x_Ocombine__in__carrier,axiom,
! [Ks2: list_list_a,Us2: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.combine_in_carrier
thf(fact_785_s_Oring_Ohom__a__inv,axiom,
! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) @ s2 )
= ( a_inv_a_b @ r @ ( eval_a_b @ r @ X3 @ s2 ) ) ) ) ).
% s.ring.hom_a_inv
thf(fact_786_subringE_I4_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( H != bot_bot_set_list_a ) ) ).
% subringE(4)
thf(fact_787_subringE_I4_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( H != bot_bot_set_a ) ) ).
% subringE(4)
thf(fact_788_empty__set,axiom,
( bot_bot_set_list_a
= ( set_list_a2 @ nil_list_a ) ) ).
% empty_set
thf(fact_789_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_790_abelian__monoid_Ofinsum__empty,axiom,
! [G: partia2175431115845679010xt_a_b,F: list_a > a] :
( ( abelian_monoid_a_b @ G )
=> ( ( finsum_a_b_list_a @ G @ F @ bot_bot_set_list_a )
= ( zero_a_b @ G ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_791_abelian__monoid_Ofinsum__empty,axiom,
! [G: partia2670972154091845814t_unit,F: list_a > list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( finsum8721804980556663006list_a @ G @ F @ bot_bot_set_list_a )
= ( zero_l4142658623432671053t_unit @ G ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_792_abelian__monoid_Ofinsum__empty,axiom,
! [G: partia2175431115845679010xt_a_b,F: a > a] :
( ( abelian_monoid_a_b @ G )
=> ( ( finsum_a_b_a @ G @ F @ bot_bot_set_a )
= ( zero_a_b @ G ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_793_abelian__monoid_Ofinsum__empty,axiom,
! [G: partia2670972154091845814t_unit,F: a > list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( finsum7322697649718157656unit_a @ G @ F @ bot_bot_set_a )
= ( zero_l4142658623432671053t_unit @ G ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_794_perm__empty__imp,axiom,
! [Ys: list_list_a] :
( ( ( mset_list_a @ nil_list_a )
= ( mset_list_a @ Ys ) )
=> ( Ys = nil_list_a ) ) ).
% perm_empty_imp
thf(fact_795_perm__empty__imp,axiom,
! [Ys: list_a] :
( ( ( mset_a @ nil_a )
= ( mset_a @ Ys ) )
=> ( Ys = nil_a ) ) ).
% perm_empty_imp
thf(fact_796_perm__sing__imp,axiom,
! [Ys: list_list_a,Xs: list_list_a,Y: list_a] :
( ( ( mset_list_a @ Ys )
= ( mset_list_a @ Xs ) )
=> ( ( Xs
= ( cons_list_a @ Y @ nil_list_a ) )
=> ( Ys
= ( cons_list_a @ Y @ nil_list_a ) ) ) ) ).
% perm_sing_imp
thf(fact_797_perm__sing__imp,axiom,
! [Ys: list_a,Xs: list_a,Y: a] :
( ( ( mset_a @ Ys )
= ( mset_a @ Xs ) )
=> ( ( Xs
= ( cons_a @ Y @ nil_a ) )
=> ( Ys
= ( cons_a @ Y @ nil_a ) ) ) ) ).
% perm_sing_imp
thf(fact_798_perm__append__single,axiom,
! [A2: list_a,Xs: list_list_a] :
( ( mset_list_a @ ( cons_list_a @ A2 @ Xs ) )
= ( mset_list_a @ ( append_list_a @ Xs @ ( cons_list_a @ A2 @ nil_list_a ) ) ) ) ).
% perm_append_single
thf(fact_799_perm__append__single,axiom,
! [A2: a,Xs: list_a] :
( ( mset_a @ ( cons_a @ A2 @ Xs ) )
= ( mset_a @ ( append_a @ Xs @ ( cons_a @ A2 @ nil_a ) ) ) ) ).
% perm_append_single
thf(fact_800_alg__multE_I1_J,axiom,
! [X3: a,P: list_a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X3 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P @ X3 ) ) @ P ) ) ) ) ).
% alg_multE(1)
thf(fact_801_x_Oconst__term__zero,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P )
=> ( ( P != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ~ ! [P4: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P4 )
=> ( ( P4 != nil_list_a )
=> ( P
!= ( append_list_a @ P4 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) ) ) ) ) ) ) ) ).
% x.const_term_zero
thf(fact_802_x_OsubcringI,axiom,
! [H: set_list_a] :
( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [H12: list_a,H22: list_a] :
( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H12 @ H22 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H22 @ H12 ) ) ) )
=> ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subcringI
thf(fact_803_x_OSpan__mem__iff__length__version,axiom,
! [K: set_list_a,Us2: list_list_a,A2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) )
= ( ? [Ks4: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks4 ) @ K )
& ( ( size_s349497388124573686list_a @ Ks4 )
= ( size_s349497388124573686list_a @ Us2 ) )
& ( A2
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks4 @ Us2 ) ) ) ) ) ) ) ).
% x.Span_mem_iff_length_version
thf(fact_804_x_Opoly__add_Ocases,axiom,
! [X3: produc7709606177366032167list_a] :
~ ! [P1: list_list_a,P22: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ P1 @ P22 ) ) ).
% x.poly_add.cases
thf(fact_805_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_806_subring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( K != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_807_x_Ocarrier__is__subcring,axiom,
subcri7763218559781929323t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.carrier_is_subcring
thf(fact_808_x_OsubcringI_H,axiom,
! [H: set_list_a] :
( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.subcringI'
thf(fact_809_add_Oone__in__subset,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H != bot_bot_set_a )
=> ( ! [X: a] :
( ( member_a @ X @ H )
=> ( member_a @ ( a_inv_a_b @ r @ X ) @ H ) )
=> ( ! [X: a] :
( ( member_a @ X @ H )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ H )
=> ( member_a @ ( add_a_b @ r @ X @ Xa2 ) @ H ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H ) ) ) ) ) ).
% add.one_in_subset
thf(fact_810_x_Opolynomial__incl,axiom,
! [K: set_list_a,P: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ K ) ) ).
% x.polynomial_incl
thf(fact_811_x_OSpan__in__carrier,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Span_in_carrier
thf(fact_812_x_Oeval__poly__in__carrier,axiom,
! [K: set_list_a,P: list_list_a,X3: list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.eval_poly_in_carrier
thf(fact_813_x_Omono__Span__subset,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) ) ) ) ) ).
% x.mono_Span_subset
thf(fact_814_x_Omono__Span__sublist,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( set_list_a2 @ Vs ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) ) ) ) ) ).
% x.mono_Span_sublist
thf(fact_815_x_OSpan__same__set,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( set_list_a2 @ Us2 )
= ( set_list_a2 @ Vs ) )
=> ( ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 )
= ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) ) ) ) ) ).
% x.Span_same_set
thf(fact_816_x_OSpan__base__incl,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ).
% x.Span_base_incl
thf(fact_817_x_OSpan__subgroup__props_I1_J,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Span_subgroup_props(1)
thf(fact_818_x_Osubalgebra__Span__incl,axiom,
! [K: set_list_a,V: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ V )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ V ) ) ) ) ).
% x.subalgebra_Span_incl
thf(fact_819_x_OSpan__subalgebraI,axiom,
! [K: set_list_a,E: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1768981623711841426t_unit @ K @ E @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ E )
=> ( ! [V3: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ V3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ V3 )
=> ( ord_le8861187494160871172list_a @ E @ V3 ) ) )
=> ( E
= ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ) ) ).
% x.Span_subalgebraI
thf(fact_820_x_OSpan__subgroup__props_I2_J,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ).
% x.Span_subgroup_props(2)
thf(fact_821_x_OSpan__subgroup__props_I3_J,axiom,
! [K: set_list_a,Us2: list_list_a,V1: list_a,V22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ V1 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) )
=> ( ( member_list_a @ V22 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ V1 @ V22 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ) ) ).
% x.Span_subgroup_props(3)
thf(fact_822_x_OSpan__smult__closed,axiom,
! [K: set_list_a,Us2: list_list_a,K3: list_a,V4: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K3 @ K )
=> ( ( member_list_a @ V4 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ V4 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ) ) ).
% x.Span_smult_closed
thf(fact_823_x_Omono__Span,axiom,
! [K: set_list_a,Us2: list_list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ U @ Us2 ) ) ) ) ) ) ).
% x.mono_Span
thf(fact_824_x_OSpan__subgroup__props_I4_J,axiom,
! [K: set_list_a,Us2: list_list_a,V4: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ V4 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ V4 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ) ).
% x.Span_subgroup_props(4)
thf(fact_825_x_Omono__Span__append_I2_J,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( append_list_a @ Vs @ Us2 ) ) ) ) ) ) ).
% x.mono_Span_append(2)
thf(fact_826_x_Omono__Span__append_I1_J,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( append_list_a @ Us2 @ Vs ) ) ) ) ) ) ).
% x.mono_Span_append(1)
thf(fact_827_x_OSpan__finite__dimension,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ).
% x.Span_finite_dimension
thf(fact_828_x_OSpan__is__subalgebra,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( embedd1768981623711841426t_unit @ K @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Span_is_subalgebra
thf(fact_829_x_Ozero__is__polynomial,axiom,
! [K: set_list_a] : ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ nil_list_a ) ).
% x.zero_is_polynomial
thf(fact_830_x_Ocarrier__polynomial,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P ) ) ) ).
% x.carrier_polynomial
thf(fact_831_x_Opolynomial__in__carrier,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.polynomial_in_carrier
thf(fact_832_subcringE_I2_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).
% subcringE(2)
thf(fact_833_subcringE_I2_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H ) ) ).
% subcringE(2)
thf(fact_834_subcringE_I7_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H2 ) @ H ) ) ) ) ).
% subcringE(7)
thf(fact_835_subcringE_I7_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( add_a_b @ R @ H1 @ H2 ) @ H ) ) ) ) ).
% subcringE(7)
thf(fact_836_subcringE_I4_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( H != bot_bot_set_list_a ) ) ).
% subcringE(4)
thf(fact_837_subcringE_I4_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( H != bot_bot_set_a ) ) ).
% subcringE(4)
thf(fact_838_subcring_Osub__m__comm,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H2 )
= ( mult_l7073676228092353617t_unit @ R @ H2 @ H1 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_839_subcring_Osub__m__comm,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( ( mult_a_ring_ext_a_b @ R @ H1 @ H2 )
= ( mult_a_ring_ext_a_b @ R @ H2 @ H1 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_840_subcringE_I6_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H2 ) @ H ) ) ) ) ).
% subcringE(6)
thf(fact_841_subcringE_I6_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H2 ) @ H ) ) ) ) ).
% subcringE(6)
thf(fact_842_subfieldE_I2_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( subcring_a_b @ K @ R ) ) ).
% subfieldE(2)
thf(fact_843_subfieldE_I2_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( subcri7763218559781929323t_unit @ K @ R ) ) ).
% subfieldE(2)
thf(fact_844_subcringE_I3_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ H ) ) ).
% subcringE(3)
thf(fact_845_subcringE_I3_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H ) ) ).
% subcringE(3)
thf(fact_846_subcringE_I5_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H3: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H3 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H3 ) @ H ) ) ) ).
% subcringE(5)
thf(fact_847_subcringE_I5_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H3: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H3 @ H )
=> ( member_a @ ( a_inv_a_b @ R @ H3 ) @ H ) ) ) ).
% subcringE(5)
thf(fact_848_subcring_Oaxioms_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( subrin6918843898125473962t_unit @ H @ R ) ) ).
% subcring.axioms(1)
thf(fact_849_subcring_Oaxioms_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( subring_a_b @ H @ R ) ) ).
% subcring.axioms(1)
thf(fact_850_univ__poly__carrier,axiom,
( polynomial_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,K5: set_a,P3: list_a] : ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K5 ) ) ) ) ) ).
% univ_poly_carrier
thf(fact_851_univ__poly__carrier,axiom,
( polyno1315193887021588240t_unit
= ( ^ [R2: partia2670972154091845814t_unit,K5: set_list_a,P3: list_list_a] : ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K5 ) ) ) ) ) ).
% univ_poly_carrier
thf(fact_852_subcringE_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subcringE(1)
thf(fact_853_subcringE_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subcringE(1)
thf(fact_854_subcringE_I1_J,axiom,
! [H: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subcri8676831449680469861t_unit @ H @ R )
=> ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% subcringE(1)
thf(fact_855_x_OsubdomainI,axiom,
! [H: set_list_a] :
( ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [H12: list_a,H22: list_a] :
( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H12 @ H22 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( H12
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
| ( H22
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) )
=> ( subdom7821232466298058046t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.subdomainI
thf(fact_856_le__alg__mult__imp__pdivides,axiom,
! [X3: a,P: list_a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P @ X3 ) )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X3 ) @ nil_a ) ) @ N ) @ P ) ) ) ) ).
% le_alg_mult_imp_pdivides
thf(fact_857_alg__multE_I2_J,axiom,
! [X3: a,P: list_a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X3 ) @ nil_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P @ X3 ) ) ) ) ) ) ).
% alg_multE(2)
thf(fact_858_x_OSpan__append__eq__set__add,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( append_list_a @ Us2 @ Vs ) )
= ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) ) ) ) ) ) ).
% x.Span_append_eq_set_add
thf(fact_859_carrier__is__subcring,axiom,
subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subcring
thf(fact_860_subcringI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subcring_a_b @ H @ r ) ) ).
% subcringI'
thf(fact_861_polynomial__incl,axiom,
! [K: set_a,P: list_a] :
( ( polynomial_a_b @ r @ K @ P )
=> ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K ) ) ).
% polynomial_incl
thf(fact_862_subcringI,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( ! [H12: a,H22: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( mult_a_ring_ext_a_b @ r @ H12 @ H22 )
= ( mult_a_ring_ext_a_b @ r @ H22 @ H12 ) ) ) )
=> ( subcring_a_b @ H @ r ) ) ) ).
% subcringI
thf(fact_863_var__closed_I2_J,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( polynomial_a_b @ r @ K @ ( var_a_b @ r ) ) ) ).
% var_closed(2)
thf(fact_864_eval__poly__in__carrier,axiom,
! [K: set_a,P: list_a,X3: a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% eval_poly_in_carrier
thf(fact_865_pdivides__iff,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( polynomial_a_b @ r @ K @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) ) ) ) ) ).
% pdivides_iff
thf(fact_866_x_Osetadd__subset__G,axiom,
! [H: set_list_a,K: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.setadd_subset_G
thf(fact_867_x_Oset__add__comm,axiom,
! [I: set_list_a,J: set_list_a] :
( ( ord_le8861187494160871172list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ J @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ J )
= ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ J @ I ) ) ) ) ).
% x.set_add_comm
thf(fact_868_x_Oset__add__closed,axiom,
! [A: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B4 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.set_add_closed
thf(fact_869_x_Osum__space__dim_I1_J,axiom,
! [K: set_list_a,E: set_list_a,F2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ F2 )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ E @ F2 ) ) ) ) ) ).
% x.sum_space_dim(1)
thf(fact_870_polynomial__pow__division,axiom,
! [P: list_a,N: nat,M2: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ M2 ) ) ) ) ).
% polynomial_pow_division
thf(fact_871_subring__polynomial__pow__division,axiom,
! [K: set_a,P: list_a,N: nat,M2: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ M2 ) ) ) ) ) ).
% subring_polynomial_pow_division
thf(fact_872_const__term__zero,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= ( zero_a_b @ r ) )
=> ~ ! [P4: list_a] :
( ( polynomial_a_b @ r @ K @ P4 )
=> ( ( P4 != nil_a )
=> ( P
!= ( append_a @ P4 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ) ) ).
% const_term_zero
thf(fact_873_zero__is__polynomial,axiom,
! [K: set_a] : ( polynomial_a_b @ r @ K @ nil_a ) ).
% zero_is_polynomial
thf(fact_874_carrier__polynomial,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( polynomial_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ P ) ) ) ).
% carrier_polynomial
thf(fact_875_polynomial__in__carrier,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% polynomial_in_carrier
thf(fact_876_one__is__polynomial,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( polynomial_a_b @ r @ K @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) ) ) ).
% one_is_polynomial
thf(fact_877_subdomainE_I2_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).
% subdomainE(2)
thf(fact_878_subdomainE_I2_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H ) ) ).
% subdomainE(2)
thf(fact_879_subdomainE_I7_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H2 ) @ H ) ) ) ) ).
% subdomainE(7)
thf(fact_880_subdomainE_I7_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( add_a_b @ R @ H1 @ H2 ) @ H ) ) ) ) ).
% subdomainE(7)
thf(fact_881_subdomainE_I4_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( H != bot_bot_set_list_a ) ) ).
% subdomainE(4)
thf(fact_882_subdomainE_I4_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( H != bot_bot_set_a ) ) ).
% subdomainE(4)
thf(fact_883_subdomainE_I6_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H2 ) @ H ) ) ) ) ).
% subdomainE(6)
thf(fact_884_subdomainE_I6_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H2 ) @ H ) ) ) ) ).
% subdomainE(6)
thf(fact_885_subdomainE_I8_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H2 )
= ( mult_l7073676228092353617t_unit @ R @ H2 @ H1 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_886_subdomainE_I8_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( ( mult_a_ring_ext_a_b @ R @ H1 @ H2 )
= ( mult_a_ring_ext_a_b @ R @ H2 @ H1 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_887_subfield_Oaxioms_I1_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( subdom7821232466298058046t_unit @ K @ R ) ) ).
% subfield.axioms(1)
thf(fact_888_subfield_Oaxioms_I1_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( subdomain_a_b @ K @ R ) ) ).
% subfield.axioms(1)
thf(fact_889_subdomainE_I3_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ H ) ) ).
% subdomainE(3)
thf(fact_890_subdomainE_I3_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H ) ) ).
% subdomainE(3)
thf(fact_891_subdomainE_I5_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H3: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H3 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H3 ) @ H ) ) ) ).
% subdomainE(5)
thf(fact_892_subdomainE_I5_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H3: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H3 @ H )
=> ( member_a @ ( a_inv_a_b @ R @ H3 ) @ H ) ) ) ).
% subdomainE(5)
thf(fact_893_subdomain_Oaxioms_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( subcri7763218559781929323t_unit @ H @ R ) ) ).
% subdomain.axioms(1)
thf(fact_894_subdomain_Oaxioms_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( subcring_a_b @ H @ R ) ) ).
% subdomain.axioms(1)
thf(fact_895_impossible__Cons,axiom,
! [Xs: list_list_a,Ys: list_list_a,X3: list_a] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs
!= ( cons_list_a @ X3 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_896_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X3: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X3 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_897_subdomainE_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subdomainE(1)
thf(fact_898_subdomainE_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subdomainE(1)
thf(fact_899_subdomainE_I1_J,axiom,
! [H: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subdom561091866123308472t_unit @ H @ R )
=> ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% subdomainE(1)
thf(fact_900_subdomain_Osubintegral,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H2 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( H1
= ( zero_l4142658623432671053t_unit @ R ) )
| ( H2
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_901_subdomain_Osubintegral,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ R @ H1 @ H2 )
= ( zero_a_b @ R ) )
=> ( ( H1
= ( zero_a_b @ R ) )
| ( H2
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_902_subdomain_Osub__one__not__zero,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_903_subdomain_Osub__one__not__zero,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_904_subdomainI,axiom,
! [H: set_a] :
( ( subcring_a_b @ H @ r )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ! [H12: a,H22: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ r @ H12 @ H22 )
= ( zero_a_b @ r ) )
=> ( ( H12
= ( zero_a_b @ r ) )
| ( H22
= ( zero_a_b @ r ) ) ) ) ) )
=> ( subdomain_a_b @ H @ r ) ) ) ) ).
% subdomainI
thf(fact_905_x_OSpan__strict__incl,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_set_list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) )
=> ? [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Vs ) )
& ~ ( member_list_a @ X @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ) ) ) ).
% x.Span_strict_incl
thf(fact_906_x_Oadd__additive__subgroups,axiom,
! [H: set_list_a,K: set_list_a] :
( ( additi4714453376129182166t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( additi4714453376129182166t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( additi4714453376129182166t_unit @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ K ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add_additive_subgroups
thf(fact_907_subdomainI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subdomain_a_b @ H @ r ) ) ).
% subdomainI'
thf(fact_908_x_OSpan__incl,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_mu3586181839180059898t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( set_list_a2 @ Us2 ) ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ).
% x.Span_incl
thf(fact_909_mult__of_Omultlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ As @ ( one_a_ring_ext_a_b @ r ) )
= ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Bs @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.multlist_perm_cong
thf(fact_910_x_Ogenideal__one,axiom,
( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.genideal_one
thf(fact_911_mult__of_Omonoid__cancel__axioms,axiom,
monoid1999574367301118026t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.monoid_cancel_axioms
thf(fact_912_mult__of_Oprime__irreducible,axiom,
! [P: a] :
( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ).
% mult_of.prime_irreducible
thf(fact_913_zero__is__prime_I2_J,axiom,
prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ ( zero_a_b @ r ) ).
% zero_is_prime(2)
thf(fact_914_zero__is__irreducible__mult,axiom,
irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( zero_a_b @ r ) ).
% zero_is_irreducible_mult
thf(fact_915_mult__of_Odivides__unit,axiom,
! [A2: a,U: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ U )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.divides_unit
thf(fact_916_mult__of_Ounit__divides,axiom,
! [U: a,A2: a] :
( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ U @ A2 ) ) ) ).
% mult_of.unit_divides
thf(fact_917_mult__of_OUnits__closed,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.Units_closed
thf(fact_918_mult__of_Odivides__trans,axiom,
! [A2: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ C ) ) ) ) ).
% mult_of.divides_trans
thf(fact_919_mult__of_Oisgcd__divides__l,axiom,
! [A2: a,B: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ A2 @ B ) ) ) ) ).
% mult_of.isgcd_divides_l
thf(fact_920_mult__of_Oisgcd__divides__r,axiom,
! [B: a,A2: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ B @ A2 )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ B @ A2 @ B ) ) ) ) ).
% mult_of.isgcd_divides_r
thf(fact_921_mult__of_Ogcdof__exists,axiom,
! [A2: a,B: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ C2 @ A2 @ B ) ) ) ) ).
% mult_of.gcdof_exists
thf(fact_922_mult__of_Oirreducible__prime,axiom,
! [P: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ( member_a @ P @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ) ).
% mult_of.irreducible_prime
thf(fact_923_divides__mult__zero,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ ( zero_a_b @ r ) )
=> ( A2
= ( zero_a_b @ r ) ) ) ) ).
% divides_mult_zero
thf(fact_924_mult__of_Oprod__unit__l,axiom,
! [A2: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.prod_unit_l
thf(fact_925_mult__of_Oprod__unit__r,axiom,
! [A2: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.prod_unit_r
thf(fact_926_mult__of_Ounit__factor,axiom,
! [A2: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.unit_factor
thf(fact_927_mult__of_Oprime__divides,axiom,
! [A2: a,B: a,P: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ A2 )
| ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ B ) ) ) ) ) ) ).
% mult_of.prime_divides
thf(fact_928_mult__of_Odivides__prod__r,axiom,
! [A2: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ).
% mult_of.divides_prod_r
thf(fact_929_mult__of_Odivides__prod__l,axiom,
! [A2: a,B: a,C: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ) ).
% mult_of.divides_prod_l
thf(fact_930_mult__of_Ol__cancel,axiom,
! [C: a,A2: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C @ A2 )
= ( mult_a_ring_ext_a_b @ r @ C @ B ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A2 = B ) ) ) ) ) ).
% mult_of.l_cancel
thf(fact_931_mult__of_Om__assoc,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X3 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% mult_of.m_assoc
thf(fact_932_mult__of_Om__comm,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X3 ) ) ) ) ).
% mult_of.m_comm
thf(fact_933_mult__of_Om__lcomm,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X3 @ Z ) ) ) ) ) ) ).
% mult_of.m_lcomm
thf(fact_934_mult__of_Or__cancel,axiom,
! [A2: a,C: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A2 @ C )
= ( mult_a_ring_ext_a_b @ r @ B @ C ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A2 = B ) ) ) ) ) ).
% mult_of.r_cancel
thf(fact_935_mult__of_Omonoid__cancelI,axiom,
( ! [A3: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A3 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B2 ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A3 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B2 @ C2 ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid1999574367301118026t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.monoid_cancelI
thf(fact_936_mult__of_Oirreducible__prodE,axiom,
! [A2: a,B: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A2 )
=> ~ ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) )
=> ~ ( ( member_a @ A2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ~ ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ B ) ) ) ) ) ) ).
% mult_of.irreducible_prodE
thf(fact_937_mult__of_Oirreducible__prod__lI,axiom,
! [B: a,A2: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ B )
=> ( ( member_a @ A2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) ) ) ) ) ) ).
% mult_of.irreducible_prod_lI
thf(fact_938_mult__of_Oirreducible__prod__rI,axiom,
! [A2: a,B: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A2 )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) ) ) ) ) ) ).
% mult_of.irreducible_prod_rI
thf(fact_939_mult__of_Ocarrier__not__empty,axiom,
( ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) )
!= bot_bot_set_a ) ).
% mult_of.carrier_not_empty
thf(fact_940_mult__of_OUnit__eq__dividesone,axiom,
! [U: a] :
( ( member_a @ U @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ U @ ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.Unit_eq_dividesone
thf(fact_941_mult__of_OUnits__inv__comm,axiom,
! [X3: a,Y: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.Units_inv_comm
thf(fact_942_mult__of_Oirrlist__perm__cong,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ As ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ X ) )
=> ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Bs ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ X4 ) ) ) ) ).
% mult_of.irrlist_perm_cong
thf(fact_943_irreducible__imp__irreducible__mult,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( irredu6211895646901577903xt_a_b @ r @ A2 )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A2 ) ) ) ).
% irreducible_imp_irreducible_mult
thf(fact_944_ring__irreducibleE_I3_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ R3 ) ) ) ).
% ring_irreducibleE(3)
thf(fact_945_ring__primeE_I2_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ) ).
% ring_primeE(2)
thf(fact_946_prime__eq__prime__mult,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P )
= ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ) ).
% prime_eq_prime_mult
thf(fact_947_mult__of_Ofactors__closed,axiom,
! [Fs: list_a,A2: a] :
( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.factors_closed
thf(fact_948_mult__of_Ofactors__dividesI,axiom,
! [Fs: list_a,A2: a,F: a] :
( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A2 )
=> ( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ F @ A2 ) ) ) ) ).
% mult_of.factors_dividesI
thf(fact_949_mult__of_Ofactors__exist,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ~ ( member_a @ A2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [Fs3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs3 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fs3 @ A2 ) ) ) ) ).
% mult_of.factors_exist
thf(fact_950_mult__of_Ofactors__mult__single,axiom,
! [A2: a,Fb: list_a,B: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A2 )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fb @ B )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ ( cons_a @ A2 @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) ) ) ) ) ).
% mult_of.factors_mult_single
thf(fact_951_mult__of_OUnits__r__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.Units_r_inv_ex
thf(fact_952_mult__of_OUnits__l__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.Units_l_inv_ex
thf(fact_953_mult__of_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X )
= X ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.one_unique
thf(fact_954_mult__of_Oinv__unique,axiom,
! [Y: a,X3: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y @ X3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% mult_of.inv_unique
thf(fact_955_mult__of_Operm__closed,axiom,
! [As: list_a,Bs: list_a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.perm_closed
thf(fact_956_mult__of_OfactorsI,axiom,
! [Fs: list_a,A2: a] :
( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Fs ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ X ) )
=> ( ( ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) )
= A2 )
=> ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A2 ) ) ) ).
% mult_of.factorsI
thf(fact_957_x_Oset__mult__closed,axiom,
! [H: set_list_a,K: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_mu3586181839180059898t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.set_mult_closed
thf(fact_958_mult__of_Ofactors__mult,axiom,
! [Fa: list_a,A2: a,Fb: list_a,B: a] :
( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fa @ A2 )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fb @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) ) ) ) ) ) ).
% mult_of.factors_mult
thf(fact_959_x_Ogenideal__self_H,axiom,
! [I2: list_a] :
( ( member_list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ I2 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I2 @ bot_bot_set_list_a ) ) ) ) ).
% x.genideal_self'
thf(fact_960_x_Ogenideal__zero,axiom,
( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ).
% x.genideal_zero
thf(fact_961_Ring__Divisibility_Omult__mult__of,axiom,
! [R: partia2670972154091845814t_unit] :
( ( mult_l6995149843440949818t_unit @ ( ring_m2863707994090333347t_unit @ R ) )
= ( mult_l7073676228092353617t_unit @ R ) ) ).
% Ring_Divisibility.mult_mult_of
thf(fact_962_Ring__Divisibility_Omult__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( mult_a_Product_unit @ ( ring_mult_of_a_b @ R ) )
= ( mult_a_ring_ext_a_b @ R ) ) ).
% Ring_Divisibility.mult_mult_of
thf(fact_963_x_Ozeropideal,axiom,
princi8786919440553033881t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.zeropideal
thf(fact_964_Ring__Divisibility_Oone__mult__of,axiom,
! [R: partia2670972154091845814t_unit] :
( ( one_li6878281577851457998t_unit @ ( ring_m2863707994090333347t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ R ) ) ).
% Ring_Divisibility.one_mult_of
thf(fact_965_Ring__Divisibility_Oone__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( one_a_Product_unit @ ( ring_mult_of_a_b @ R ) )
= ( one_a_ring_ext_a_b @ R ) ) ).
% Ring_Divisibility.one_mult_of
thf(fact_966_x_Ocarrier__one__not__zero,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.carrier_one_not_zero
thf(fact_967_x_Ocarrier__one__zero,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.carrier_one_zero
thf(fact_968_x_Oone__zeroD,axiom,
( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ).
% x.one_zeroD
thf(fact_969_x_Oone__zeroI,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
=> ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.one_zeroI
thf(fact_970_x_OIdl__subset__ideal_H,axiom,
! [A2: list_a,B: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) )
= ( member_list_a @ A2 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ) ) ).
% x.Idl_subset_ideal'
thf(fact_971_mult__of_Omultlist__dividesI,axiom,
! [F: a,Fs: list_a] :
( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ F @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.multlist_dividesI
thf(fact_972_x_OSpan_Oelims,axiom,
! [X3: set_list_a,Xa3: list_list_a,Y: set_list_a] :
( ( ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Xa3 )
= Y )
=> ( Y
= ( foldr_4630022767072968199list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) @ Xa3 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ).
% x.Span.elims
thf(fact_973_x_OSpan_Osimps,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 )
= ( foldr_4630022767072968199list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ Us2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ).
% x.Span.simps
thf(fact_974_mult__of_Olcmof__exists,axiom,
! [A2: a,B: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( islcm_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ C2 @ A2 @ B ) ) ) ) ).
% mult_of.lcmof_exists
thf(fact_975_mult__of_Odivides__refl,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ A2 ) ) ).
% mult_of.divides_refl
thf(fact_976_mult__of_OUnits__m__closed,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.Units_m_closed
thf(fact_977_mult__of_OUnits__one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ).
% mult_of.Units_one_closed
thf(fact_978_Units__mult__eq__Units,axiom,
( ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) )
= ( units_a_ring_ext_a_b @ r ) ) ).
% Units_mult_eq_Units
thf(fact_979_mult__of_OSomeGcd__ex,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( A != bot_bot_set_a )
=> ( member_a @ ( someGc8133249837406473920t_unit @ ( ring_mult_of_a_b @ r ) @ A ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.SomeGcd_ex
thf(fact_980_mult__of_OUnits__l__cancel,axiom,
! [X3: a,Y: a,Z: a] :
( ( member_a @ X3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y )
= ( mult_a_ring_ext_a_b @ r @ X3 @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% mult_of.Units_l_cancel
thf(fact_981_mult__of_Odivides__mult__rI,axiom,
! [A2: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% mult_of.divides_mult_rI
thf(fact_982_mult__of_Odivides__mult__r,axiom,
! [A2: a,B: a,C: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B ) ) ) ) ) ).
% mult_of.divides_mult_r
thf(fact_983_mult__of_Odivides__mult__lI,axiom,
! [A2: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ A2 ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% mult_of.divides_mult_lI
thf(fact_984_mult__of_Odivides__mult__l,axiom,
! [A2: a,B: a,C: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ A2 ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B ) ) ) ) ) ).
% mult_of.divides_mult_l
thf(fact_985_mult__of_Om__closed,axiom,
! [X3: a,Y: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.m_closed
thf(fact_986_mult__of_Oone__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ).
% mult_of.one_closed
thf(fact_987_mult__of_Or__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( one_a_ring_ext_a_b @ r ) )
= X3 ) ) ).
% mult_of.r_one
thf(fact_988_mult__of_Ol__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X3 )
= X3 ) ) ).
% mult_of.l_one
thf(fact_989_mult__of_Odivisor__chain__condition__monoid__axioms,axiom,
diviso6259607970152342594t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.divisor_chain_condition_monoid_axioms
thf(fact_990_mult__of_Omultlist__closed,axiom,
! [Fs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.multlist_closed
thf(fact_991_mult__of_Oprimeness__condition__monoid__axioms,axiom,
primen965786292471834261t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.primeness_condition_monoid_axioms
thf(fact_992_divides__mult__imp__divides,axiom,
! [R: partia2670972154091845814t_unit,A2: list_a,B: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ R ) @ A2 @ B )
=> ( factor1757716651909850160t_unit @ R @ A2 @ B ) ) ).
% divides_mult_imp_divides
thf(fact_993_divides__mult__imp__divides,axiom,
! [R: partia2956882679547061052t_unit,A2: list_list_a,B: list_list_a] :
( ( factor5444141823398166165t_unit @ ( ring_m2698952956457993885t_unit @ R ) @ A2 @ B )
=> ( factor6954119973539764400t_unit @ R @ A2 @ B ) ) ).
% divides_mult_imp_divides
thf(fact_994_divides__mult__imp__divides,axiom,
! [R: partia2175431115845679010xt_a_b,A2: a,B: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ R ) @ A2 @ B )
=> ( factor8216151070175719842xt_a_b @ R @ A2 @ B ) ) ).
% divides_mult_imp_divides
thf(fact_995_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
!= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_996_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_997_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( ( partia2464479390973590831t_unit @ R )
!= ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) )
= ( ( one_li8234411390022467901t_unit @ R )
!= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_998_semiring_Ocarrier__one__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_999_semiring_Ocarrier__one__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_1000_semiring_Ocarrier__one__zero,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( ( partia2464479390973590831t_unit @ R )
= ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) )
= ( ( one_li8234411390022467901t_unit @ R )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_1001_semiring_Oone__zeroI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_1002_semiring_Oone__zeroI,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
=> ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_1003_semiring_Oone__zeroI,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( ( partia2464479390973590831t_unit @ R )
= ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) )
=> ( ( one_li8234411390022467901t_unit @ R )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_1004_semiring_Oone__zeroD,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) )
=> ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_1005_semiring_Oone__zeroD,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_1006_semiring_Oone__zeroD,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( ( one_li8234411390022467901t_unit @ R )
= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( partia2464479390973590831t_unit @ R )
= ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_1007_mult__of_Odivides__fcount,axiom,
! [A2: a,B: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ord_less_eq_nat @ ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ A2 ) @ ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ B ) ) ) ) ) ).
% mult_of.divides_fcount
thf(fact_1008_x_Ozeromaximalideal__eq__field,axiom,
( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.zeromaximalideal_eq_field
thf(fact_1009_x_Ozeromaximalideal__fieldI,axiom,
( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.zeromaximalideal_fieldI
thf(fact_1010_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_1011_one__zeroI,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% one_zeroI
thf(fact_1012_one__zeroD,axiom,
( ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) )
=> ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% one_zeroD
thf(fact_1013_carrier__one__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% carrier_one_zero
thf(fact_1014_carrier__one__not__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) ) ) ).
% carrier_one_not_zero
thf(fact_1015_islcm__def,axiom,
( islcm_a_ring_ext_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b,X2: a,A4: a,B3: a] :
( ( factor8216151070175719842xt_a_b @ G2 @ A4 @ X2 )
& ( factor8216151070175719842xt_a_b @ G2 @ B3 @ X2 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( ( factor8216151070175719842xt_a_b @ G2 @ A4 @ Y4 )
& ( factor8216151070175719842xt_a_b @ G2 @ B3 @ Y4 ) )
=> ( factor8216151070175719842xt_a_b @ G2 @ X2 @ Y4 ) ) ) ) ) ) ).
% islcm_def
thf(fact_1016_islcm__def,axiom,
( islcm_2868663726700424025t_unit
= ( ^ [G2: partia2670972154091845814t_unit,X2: list_a,A4: list_a,B3: list_a] :
( ( factor1757716651909850160t_unit @ G2 @ A4 @ X2 )
& ( factor1757716651909850160t_unit @ G2 @ B3 @ X2 )
& ! [Y4: list_a] :
( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( ( factor1757716651909850160t_unit @ G2 @ A4 @ Y4 )
& ( factor1757716651909850160t_unit @ G2 @ B3 @ Y4 ) )
=> ( factor1757716651909850160t_unit @ G2 @ X2 @ Y4 ) ) ) ) ) ) ).
% islcm_def
thf(fact_1017_islcm__def,axiom,
( islcm_8506699153953918681t_unit
= ( ^ [G2: partia2956882679547061052t_unit,X2: list_list_a,A4: list_list_a,B3: list_list_a] :
( ( factor6954119973539764400t_unit @ G2 @ A4 @ X2 )
& ( factor6954119973539764400t_unit @ G2 @ B3 @ X2 )
& ! [Y4: list_list_a] :
( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( ( factor6954119973539764400t_unit @ G2 @ A4 @ Y4 )
& ( factor6954119973539764400t_unit @ G2 @ B3 @ Y4 ) )
=> ( factor6954119973539764400t_unit @ G2 @ X2 @ Y4 ) ) ) ) ) ) ).
% islcm_def
thf(fact_1018_islcm__def,axiom,
( islcm_a_Product_unit
= ( ^ [G2: partia8223610829204095565t_unit,X2: a,A4: a,B3: a] :
( ( factor3040189038382604065t_unit @ G2 @ A4 @ X2 )
& ( factor3040189038382604065t_unit @ G2 @ B3 @ X2 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( partia6735698275553448452t_unit @ G2 ) )
=> ( ( ( factor3040189038382604065t_unit @ G2 @ A4 @ Y4 )
& ( factor3040189038382604065t_unit @ G2 @ B3 @ Y4 ) )
=> ( factor3040189038382604065t_unit @ G2 @ X2 @ Y4 ) ) ) ) ) ) ).
% islcm_def
thf(fact_1019_primeness__condition__monoid_Oirreducible__prime,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a] :
( ( primen9005823089519874350xt_a_b @ G )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( irredu6211895646901577903xt_a_b @ G @ A2 )
=> ( prime_a_ring_ext_a_b @ G @ A2 ) ) ) ) ).
% primeness_condition_monoid.irreducible_prime
thf(fact_1020_primeness__condition__monoid_Oirreducible__prime,axiom,
! [G: partia2670972154091845814t_unit,A2: list_a] :
( ( primen273420296629541820t_unit @ G )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( irredu4230924414530676029t_unit @ G @ A2 )
=> ( prime_2011924034616061926t_unit @ G @ A2 ) ) ) ) ).
% primeness_condition_monoid.irreducible_prime
thf(fact_1021_primeness__condition__monoid_Oirreducible__prime,axiom,
! [G: partia2956882679547061052t_unit,A2: list_list_a] :
( ( primen3532642134360990780t_unit @ G )
=> ( ( member_list_list_a @ A2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( irredu4439051761327310013t_unit @ G @ A2 )
=> ( prime_1232919612140715622t_unit @ G @ A2 ) ) ) ) ).
% primeness_condition_monoid.irreducible_prime
thf(fact_1022_primeness__condition__monoid_Oirreducible__prime,axiom,
! [G: partia8223610829204095565t_unit,A2: a] :
( ( primen965786292471834261t_unit @ G )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( irredu4023057619401689684t_unit @ G @ A2 )
=> ( prime_a_Product_unit @ G @ A2 ) ) ) ) ).
% primeness_condition_monoid.irreducible_prime
thf(fact_1023_zeromaximalideal__fieldI,axiom,
( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
=> ( field_a_b @ r ) ) ).
% zeromaximalideal_fieldI
thf(fact_1024_zeromaximalideal__eq__field,axiom,
( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
= ( field_a_b @ r ) ) ).
% zeromaximalideal_eq_field
thf(fact_1025_x_OSpan__mem__imp__non__trivial__combine,axiom,
! [K: set_list_a,Us2: list_list_a,A2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) )
=> ~ ! [K4: list_a] :
( ( member_list_a @ K4 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ! [Ks: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks ) @ K )
=> ( ( ( size_s349497388124573686list_a @ Ks )
= ( size_s349497388124573686list_a @ Us2 ) )
=> ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ K4 @ Ks ) @ ( cons_list_a @ A2 @ Us2 ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ) ).
% x.Span_mem_imp_non_trivial_combine
thf(fact_1026_zeromaximalideal,axiom,
maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeromaximalideal
thf(fact_1027_x_Olead__coeff__not__zero,axiom,
! [K: set_list_a,A2: list_a,P: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ A2 @ P ) )
=> ( member_list_a @ A2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ).
% x.lead_coeff_not_zero
thf(fact_1028_x_Osubfield__m__inv__simprule,axiom,
! [K: set_list_a,K3: list_a,A2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K3 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ A2 ) @ K )
=> ( member_list_a @ A2 @ K ) ) ) ) ) ).
% x.subfield_m_inv_simprule
thf(fact_1029_x_Ocring__fieldI,axiom,
( ( ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.cring_fieldI
thf(fact_1030_x_Olead__coeff__in__carrier,axiom,
! [K: set_list_a,A2: list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ A2 @ P ) )
=> ( member_list_a @ A2 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ) ).
% x.lead_coeff_in_carrier
thf(fact_1031_x_Ofield__intro2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.field_intro2
thf(fact_1032_x_OSpan__m__inv__simprule,axiom,
! [K: set_list_a,Us2: list_list_a,K3: list_a,A2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K3 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ A2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) )
=> ( member_list_a @ A2 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ) ) ) ).
% x.Span_m_inv_simprule
thf(fact_1033_x_OSpan__mem__iff,axiom,
! [K: set_list_a,Us2: list_list_a,A2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
& ? [Ks4: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks4 ) @ K )
& ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ X2 @ Ks4 ) @ ( cons_list_a @ A2 @ Us2 ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ) ) ).
% x.Span_mem_iff
thf(fact_1034_Ring__Divisibility_Ocarrier__mult__of,axiom,
! [R: partia2670972154091845814t_unit] :
( ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ R ) )
= ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ).
% Ring_Divisibility.carrier_mult_of
thf(fact_1035_Ring__Divisibility_Ocarrier__mult__of,axiom,
! [R: partia2956882679547061052t_unit] :
( ( partia7172811403827572716t_unit @ ( ring_m2698952956457993885t_unit @ R ) )
= ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) ) ).
% Ring_Divisibility.carrier_mult_of
thf(fact_1036_Ring__Divisibility_Ocarrier__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ R ) )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ).
% Ring_Divisibility.carrier_mult_of
thf(fact_1037_x_Oconst__is__polynomial,axiom,
! [A2: list_a,K: set_list_a] :
( ( member_list_a @ A2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ A2 @ nil_list_a ) ) ) ).
% x.const_is_polynomial
thf(fact_1038_x_Omonom__is__polynomial,axiom,
! [K: set_list_a,A2: list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ A2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ N ) ) ) ) ).
% x.monom_is_polynomial
thf(fact_1039_Ring_Ofield__Units,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( ( units_5837875185506529638t_unit @ R )
= ( minus_4782336368215558443list_a @ ( partia141011252114345353t_unit @ R ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) ) ) ).
% Ring.field_Units
thf(fact_1040_Ring_Ofield__Units,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( units_a_ring_ext_a_b @ R )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ).
% Ring.field_Units
thf(fact_1041_Ring_Ofield__Units,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( units_2932844235741507942t_unit @ R )
= ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ) ).
% Ring.field_Units
thf(fact_1042_Ring_Ofield__Units,axiom,
! [R: partia2956882679547061052t_unit] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( units_4903515905731149798t_unit @ R )
= ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) ) ) ).
% Ring.field_Units
thf(fact_1043_noetherian__domain_Oexists__irreducible__divisor,axiom,
! [R: partia2670972154091845814t_unit,A2: list_a] :
( ( ring_n4705423059119889713t_unit @ R )
=> ( ( member_list_a @ A2 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ~ ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ R ) )
=> ~ ! [B2: list_a] :
( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ B2 )
=> ~ ( factor1757716651909850160t_unit @ R @ B2 @ A2 ) ) ) ) ) ) ).
% noetherian_domain.exists_irreducible_divisor
thf(fact_1044_noetherian__domain_Oexists__irreducible__divisor,axiom,
! [R: partia2956882679547061052t_unit,A2: list_list_a] :
( ( ring_n8900817365880610859t_unit @ R )
=> ( ( member_list_list_a @ A2 @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ~ ( member_list_list_a @ A2 @ ( units_4903515905731149798t_unit @ R ) )
=> ~ ! [B2: list_list_a] :
( ( member_list_list_a @ B2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ B2 )
=> ~ ( factor6954119973539764400t_unit @ R @ B2 @ A2 ) ) ) ) ) ) ).
% noetherian_domain.exists_irreducible_divisor
thf(fact_1045_noetherian__domain_Oexists__irreducible__divisor,axiom,
! [R: partia2175431115845679010xt_a_b,A2: a] :
( ( ring_n4045954140777738665in_a_b @ R )
=> ( ( member_a @ A2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ A2 @ ( units_a_ring_ext_a_b @ R ) )
=> ~ ! [B2: a] :
( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ B2 )
=> ~ ( factor8216151070175719842xt_a_b @ R @ B2 @ A2 ) ) ) ) ) ) ).
% noetherian_domain.exists_irreducible_divisor
thf(fact_1046_monom__eq__var__pow,axiom,
! [K: set_a,A2: a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( monom_a_b @ r @ A2 @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ A2 @ nil_a ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) ) ) ) ) ).
% monom_eq_var_pow
thf(fact_1047_x_Opoly__add__monom,axiom,
! [P: list_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ ( size_s349497388124573686list_a @ P ) ) @ P )
= ( cons_list_a @ A2 @ P ) ) ) ) ).
% x.poly_add_monom
thf(fact_1048_x_Opoly__add__closed,axiom,
! [K: set_list_a,P12: list_list_a,P23: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P12 )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P23 )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P12 @ P23 ) ) ) ) ) ).
% x.poly_add_closed
thf(fact_1049_x_Opoly__add__comm,axiom,
! [P12: list_list_a,P23: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P12 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P23 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P12 @ P23 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P23 @ P12 ) ) ) ) ).
% x.poly_add_comm
thf(fact_1050_x_Opoly__add__in__carrier,axiom,
! [P12: list_list_a,P23: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P12 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P23 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P12 @ P23 ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.poly_add_in_carrier
thf(fact_1051_local_Ofield__Units,axiom,
( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% local.field_Units
thf(fact_1052_lead__coeff__not__zero,axiom,
! [K: set_a,A2: a,P: list_a] :
( ( polynomial_a_b @ r @ K @ ( cons_a @ A2 @ P ) )
=> ( member_a @ A2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% lead_coeff_not_zero
thf(fact_1053_x_Opoly__add__zero_I1_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ nil_list_a )
= P ) ) ) ).
% x.poly_add_zero(1)
thf(fact_1054_x_Opoly__add__zero_I2_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ P )
= P ) ) ) ).
% x.poly_add_zero(2)
thf(fact_1055_subfield__m__inv__simprule,axiom,
! [K: set_a,K3: a,A2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K3 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ A2 ) @ K )
=> ( member_a @ A2 @ K ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_1056_mult__divides,axiom,
! [A2: a,B: a,C: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A2 ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) )
=> ( factor8216151070175719842xt_a_b @ r @ A2 @ B ) ) ) ) ) ).
% mult_divides
thf(fact_1057_x_Opoly__add__is__polynomial,axiom,
! [K: set_list_a,P12: list_list_a,P23: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P12 ) @ K )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P23 ) @ K )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P12 @ P23 ) ) ) ) ) ).
% x.poly_add_is_polynomial
thf(fact_1058_cring__fieldI,axiom,
( ( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( field_a_b @ r ) ) ).
% cring_fieldI
thf(fact_1059_x_Oeval__poly__add,axiom,
! [P: list_list_a,Q: list_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ A2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A2 ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ A2 ) ) ) ) ) ) ).
% x.eval_poly_add
thf(fact_1060_x_Oconst__term__simprules_I3_J,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q ) ) ) ) ) ).
% x.const_term_simprules(3)
thf(fact_1061_divides__imp__divides__mult,axiom,
! [A2: a,B: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A2 @ B )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B ) ) ) ) ).
% divides_imp_divides_mult
thf(fact_1062_lead__coeff__in__carrier,axiom,
! [K: set_a,A2: a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ ( cons_a @ A2 @ P ) )
=> ( member_a @ A2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ).
% lead_coeff_in_carrier
thf(fact_1063_ring__irreducibleI,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R3
= ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ r @ R3 ) ) ) ) ).
% ring_irreducibleI
thf(fact_1064_irreducible__mult__imp__irreducible,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A2 )
=> ( irredu6211895646901577903xt_a_b @ r @ A2 ) ) ) ).
% irreducible_mult_imp_irreducible
thf(fact_1065_field__intro2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) ) )
=> ( field_a_b @ r ) ) ) ).
% field_intro2
thf(fact_1066_ring__irreducibleI_H,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ R3 )
=> ( ring_r999134135267193926le_a_b @ r @ R3 ) ) ) ).
% ring_irreducibleI'
thf(fact_1067_exists__irreducible__divisor,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) )
=> ~ ! [B2: a] :
( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ B2 )
=> ~ ( factor8216151070175719842xt_a_b @ r @ B2 @ A2 ) ) ) ) ) ).
% exists_irreducible_divisor
thf(fact_1068_ring__primeI_H,axiom,
! [P: a] :
( ( member_a @ P @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% ring_primeI'
thf(fact_1069_x_Oeval__poly__add__aux,axiom,
! [P: list_list_a,Q: list_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( size_s349497388124573686list_a @ P )
= ( size_s349497388124573686list_a @ Q ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ A2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A2 ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ A2 ) ) ) ) ) ) ) ).
% x.eval_poly_add_aux
thf(fact_1070_const__is__polynomial,axiom,
! [A2: a,K: set_a] :
( ( member_a @ A2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( polynomial_a_b @ r @ K @ ( cons_a @ A2 @ nil_a ) ) ) ).
% const_is_polynomial
thf(fact_1071_monom__is__polynomial,axiom,
! [K: set_a,A2: a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( polynomial_a_b @ r @ K @ ( monom_a_b @ r @ A2 @ N ) ) ) ) ).
% monom_is_polynomial
thf(fact_1072_ring_Opoly__add_Ocong,axiom,
poly_a7601779127272115787t_unit = poly_a7601779127272115787t_unit ).
% ring.poly_add.cong
thf(fact_1073_ring_Opoly__add_Ocong,axiom,
poly_add_a_b = poly_add_a_b ).
% ring.poly_add.cong
thf(fact_1074_univ__poly__add,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) )
= ( poly_a7601779127272115787t_unit @ R ) ) ).
% univ_poly_add
thf(fact_1075_univ__poly__add,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) )
= ( poly_add_a_b @ R ) ) ).
% univ_poly_add
thf(fact_1076_x_Odependent__imp__non__trivial__combine,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ~ ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 )
=> ~ ! [Ks: list_list_a] :
( ( ( size_s349497388124573686list_a @ Ks )
= ( size_s349497388124573686list_a @ Us2 ) )
=> ( ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ Us2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks ) @ K )
=> ( ( set_list_a2 @ Ks )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ) ) ) ) ).
% x.dependent_imp_non_trivial_combine
thf(fact_1077_associated__polynomials__iff,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
& ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ X2 @ nil_a ) @ Q ) ) ) ) ) ) ) ) ).
% associated_polynomials_iff
thf(fact_1078_poly__add__closed,axiom,
! [K: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P12 )
=> ( ( polynomial_a_b @ r @ K @ P23 )
=> ( polynomial_a_b @ r @ K @ ( poly_add_a_b @ r @ P12 @ P23 ) ) ) ) ) ).
% poly_add_closed
thf(fact_1079_poly__add__in__carrier,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_add_a_b @ r @ P12 @ P23 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_add_in_carrier
thf(fact_1080_poly__add__comm,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P12 @ P23 )
= ( poly_add_a_b @ r @ P23 @ P12 ) ) ) ) ).
% poly_add_comm
thf(fact_1081_x_Oassociated__sym,axiom,
! [A2: list_a,B: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A2 ) ) ).
% x.associated_sym
thf(fact_1082_poly__add__zero_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_add_a_b @ r @ P @ nil_a )
= P ) ) ) ).
% poly_add_zero(1)
thf(fact_1083_poly__add__zero_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_add_a_b @ r @ nil_a @ P )
= P ) ) ) ).
% poly_add_zero(2)
thf(fact_1084_x_Oassociated__trans,axiom,
! [A2: list_a,B: list_a,C: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ C ) ) ) ) ) ).
% x.associated_trans
thf(fact_1085_x_Oassoc__subst,axiom,
! [A2: list_a,B: list_a,F: list_a > list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B )
=> ( ! [A3: list_a,B2: list_a] :
( ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ B2 ) )
=> ( ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( member_list_a @ ( F @ B2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ A3 ) @ ( F @ B2 ) ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ A2 ) @ ( F @ B ) ) ) ) ) ) ).
% x.assoc_subst
thf(fact_1086_x_Oindependent__backwards_I2_J,axiom,
! [K: set_list_a,U: list_a,Us2: list_list_a] :
( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ U @ Us2 ) )
=> ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ).
% x.independent_backwards(2)
thf(fact_1087_x_Oli__Nil,axiom,
! [K: set_list_a] : ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ nil_list_a ) ).
% x.li_Nil
thf(fact_1088_x_OUnits__assoc,axiom,
! [A2: list_a,B: list_a] :
( ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) ) ) ).
% x.Units_assoc
thf(fact_1089_poly__add__is__polynomial,axiom,
! [K: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ K )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ K )
=> ( polynomial_a_b @ r @ K @ ( poly_add_a_b @ r @ P12 @ P23 ) ) ) ) ) ).
% poly_add_is_polynomial
thf(fact_1090_eval__poly__add,axiom,
! [P: list_a,Q: list_a,A2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) @ A2 )
= ( add_a_b @ r @ ( eval_a_b @ r @ P @ A2 ) @ ( eval_a_b @ r @ Q @ A2 ) ) ) ) ) ) ).
% eval_poly_add
thf(fact_1091_x_Omult__cong__r,axiom,
! [B: list_a,B5: list_a,A2: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ B5 )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B5 ) ) ) ) ) ) ).
% x.mult_cong_r
thf(fact_1092_x_Omult__cong__l,axiom,
! [A2: list_a,A5: list_a,B: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ A5 )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A5 @ B ) ) ) ) ) ) ).
% x.mult_cong_l
thf(fact_1093_x_Oindependent__backwards_I3_J,axiom,
! [K: set_list_a,U: list_a,Us2: list_list_a] :
( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ U @ Us2 ) )
=> ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.independent_backwards(3)
thf(fact_1094_x_OUnits__cong,axiom,
! [A2: list_a,B: list_a] :
( ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.Units_cong
thf(fact_1095_x_Odivides__cong__l,axiom,
! [X3: list_a,X5: list_a,Y: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ X5 )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X5 @ Y )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y ) ) ) ) ).
% x.divides_cong_l
thf(fact_1096_x_Odivides__cong__r,axiom,
! [X3: list_a,Y: list_a,Y2: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Y2 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Y2 ) ) ) ) ).
% x.divides_cong_r
thf(fact_1097_associated__polynomials__imp__same__length,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
=> ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q ) ) ) ) ) ) ).
% associated_polynomials_imp_same_length
thf(fact_1098_const__term__simprules_I3_J,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) )
= ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ).
% const_term_simprules(3)
thf(fact_1099_x_Oindependent__backwards_I1_J,axiom,
! [K: set_list_a,U: list_a,Us2: list_list_a] :
( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ U @ Us2 ) )
=> ~ ( member_list_a @ U @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ).
% x.independent_backwards(1)
thf(fact_1100_x_Oindependent__split_I2_J,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( append_list_a @ Us2 @ Vs ) )
=> ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ).
% x.independent_split(2)
thf(fact_1101_x_Oindependent__split_I1_J,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( append_list_a @ Us2 @ Vs ) )
=> ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) ) ) ).
% x.independent_split(1)
thf(fact_1102_associated__polynomials__imp__same__is__root,axiom,
! [P: list_a,Q: list_a,X3: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X3 )
= ( polyno4133073214067823460ot_a_b @ r @ Q @ X3 ) ) ) ) ) ).
% associated_polynomials_imp_same_is_root
thf(fact_1103_eval__poly__add__aux,axiom,
! [P: list_a,Q: list_a,A2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) @ A2 )
= ( add_a_b @ r @ ( eval_a_b @ r @ P @ A2 ) @ ( eval_a_b @ r @ Q @ A2 ) ) ) ) ) ) ) ).
% eval_poly_add_aux
thf(fact_1104_x_Oindependent__in__carrier,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.independent_in_carrier
thf(fact_1105_x_OassociatedI2,axiom,
! [U: list_a,A2: list_a,B: list_a] :
( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ U ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) ) ) ) ).
% x.associatedI2
thf(fact_1106_x_OassociatedI2_H,axiom,
! [A2: list_a,B: list_a,U: list_a] :
( ( A2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ U ) )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) ) ) ) ).
% x.associatedI2'
thf(fact_1107_x_Oli__Cons,axiom,
! [U: list_a,K: set_list_a,Us2: list_list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ~ ( member_list_a @ U @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 )
=> ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ U @ Us2 ) ) ) ) ) ).
% x.li_Cons
thf(fact_1108_x_Oindependent__same__set,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( set_list_a2 @ Us2 )
= ( set_list_a2 @ Vs ) )
=> ( ( ( size_s349497388124573686list_a @ Us2 )
= ( size_s349497388124573686list_a @ Vs ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 )
=> ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) ) ) ) ) ).
% x.independent_same_set
thf(fact_1109_divides__pirreducible__condition,axiom,
! [K: set_a,Q: list_a,P: list_a] :
( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ Q )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
=> ( ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) ) ) ) ) ).
% divides_pirreducible_condition
thf(fact_1110_x_Oindependent_Ocases,axiom,
! [A1: set_list_a,A22: list_list_a] :
( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A1 @ A22 )
=> ( ( A22 != nil_list_a )
=> ~ ! [U2: list_a,Us: list_list_a] :
( ( A22
= ( cons_list_a @ U2 @ Us ) )
=> ( ( member_list_a @ U2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ~ ( member_list_a @ U2 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A1 @ Us ) )
=> ~ ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A1 @ Us ) ) ) ) ) ) ).
% x.independent.cases
thf(fact_1111_x_Oindependent_Osimps,axiom,
! [A1: set_list_a,A22: list_list_a] :
( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A1 @ A22 )
= ( ? [K5: set_list_a] :
( ( A1 = K5 )
& ( A22 = nil_list_a ) )
| ? [U3: list_a,K5: set_list_a,Us3: list_list_a] :
( ( A1 = K5 )
& ( A22
= ( cons_list_a @ U3 @ Us3 ) )
& ( member_list_a @ U3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ~ ( member_list_a @ U3 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K5 @ Us3 ) )
& ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K5 @ Us3 ) ) ) ) ).
% x.independent.simps
thf(fact_1112_subring__degree__one__associatedI,axiom,
! [K: set_a,A2: a,A5: a,B: a] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A2 @ K )
=> ( ( member_a @ A5 @ K )
=> ( ( member_a @ B @ K )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A2 @ A5 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ A2 @ ( cons_a @ B @ nil_a ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( mult_a_ring_ext_a_b @ r @ A5 @ B ) @ nil_a ) ) ) ) ) ) ) ) ).
% subring_degree_one_associatedI
thf(fact_1113_x_Oindependent__rotate1__aux,axiom,
! [K: set_list_a,U: list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ U @ ( append_list_a @ Us2 @ Vs ) ) )
=> ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( append_list_a @ ( append_list_a @ Us2 @ ( cons_list_a @ U @ nil_list_a ) ) @ Vs ) ) ) ) ).
% x.independent_rotate1_aux
thf(fact_1114_x_Oindependent__strict__incl,axiom,
! [K: set_list_a,U: list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ U @ Us2 ) )
=> ( ord_less_set_list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ U @ Us2 ) ) ) ) ) ).
% x.independent_strict_incl
thf(fact_1115_x_Ofilter__base,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ~ ! [Vs2: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs2 )
=> ( ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs2 )
!= ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ) ) ).
% x.filter_base
thf(fact_1116_x_Oindependent__replacement,axiom,
! [K: set_list_a,U: list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ U @ Us2 ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs )
=> ( ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ U @ Us2 ) ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) )
=> ? [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Vs ) )
& ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ X @ Us2 ) ) ) ) ) ) ) ).
% x.independent_replacement
thf(fact_1117_x_Oindependent__length__le,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) )
=> ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Us2 ) @ ( size_s349497388124573686list_a @ Vs ) ) ) ) ) ) ).
% x.independent_length_le
thf(fact_1118_x_Oreplacement__theorem,axiom,
! [K: set_list_a,Us4: list_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( append_list_a @ Us4 @ Us2 ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs )
=> ( ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( append_list_a @ Us4 @ Us2 ) ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) )
=> ? [Vs3: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs3 ) @ ( set_list_a2 @ Vs ) )
& ( ( size_s349497388124573686list_a @ Vs3 )
= ( size_s349497388124573686list_a @ Us4 ) )
& ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( append_list_a @ Vs3 @ Us2 ) ) ) ) ) ) ) ).
% x.replacement_theorem
thf(fact_1119_x_Ounique__decomposition,axiom,
! [K: set_list_a,Us2: list_list_a,A2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 )
=> ( ( member_list_a @ A2 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) )
=> ? [X: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ X ) @ K )
& ( ( size_s349497388124573686list_a @ X )
= ( size_s349497388124573686list_a @ Us2 ) )
& ( A2
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Us2 ) )
& ! [Y3: list_list_a] :
( ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Y3 ) @ K )
& ( ( size_s349497388124573686list_a @ Y3 )
= ( size_s349497388124573686list_a @ Us2 ) )
& ( A2
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y3 @ Us2 ) ) )
=> ( Y3 = X ) ) ) ) ) ) ).
% x.unique_decomposition
thf(fact_1120_poly__add__monom,axiom,
! [P: list_a,A2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( poly_add_a_b @ r @ ( monom_a_b @ r @ A2 @ ( size_size_list_a @ P ) ) @ P )
= ( cons_a @ A2 @ P ) ) ) ) ).
% poly_add_monom
thf(fact_1121_x_Oassociated__refl,axiom,
! [A2: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ A2 ) ) ).
% x.associated_refl
thf(fact_1122_divides__antisym,axiom,
! [G: partia2670972154091845814t_unit,A2: list_a,B: list_a] :
( ( factor1757716651909850160t_unit @ G @ A2 @ B )
=> ( ( factor1757716651909850160t_unit @ G @ B @ A2 )
=> ( associ8407585678920448409t_unit @ G @ A2 @ B ) ) ) ).
% divides_antisym
thf(fact_1123_divides__antisym,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B: a] :
( ( factor8216151070175719842xt_a_b @ G @ A2 @ B )
=> ( ( factor8216151070175719842xt_a_b @ G @ B @ A2 )
=> ( associ5860276527279195403xt_a_b @ G @ A2 @ B ) ) ) ).
% divides_antisym
thf(fact_1124_divides__antisym,axiom,
! [G: partia8223610829204095565t_unit,A2: a,B: a] :
( ( factor3040189038382604065t_unit @ G @ A2 @ B )
=> ( ( factor3040189038382604065t_unit @ G @ B @ A2 )
=> ( associ6879500422977059064t_unit @ G @ A2 @ B ) ) ) ).
% divides_antisym
thf(fact_1125_divides__antisym,axiom,
! [G: partia2956882679547061052t_unit,A2: list_list_a,B: list_list_a] :
( ( factor6954119973539764400t_unit @ G @ A2 @ B )
=> ( ( factor6954119973539764400t_unit @ G @ B @ A2 )
=> ( associ5603075271488036121t_unit @ G @ A2 @ B ) ) ) ).
% divides_antisym
thf(fact_1126_associated__def,axiom,
( associ8407585678920448409t_unit
= ( ^ [G2: partia2670972154091845814t_unit,A4: list_a,B3: list_a] :
( ( factor1757716651909850160t_unit @ G2 @ A4 @ B3 )
& ( factor1757716651909850160t_unit @ G2 @ B3 @ A4 ) ) ) ) ).
% associated_def
thf(fact_1127_associated__def,axiom,
( associ5860276527279195403xt_a_b
= ( ^ [G2: partia2175431115845679010xt_a_b,A4: a,B3: a] :
( ( factor8216151070175719842xt_a_b @ G2 @ A4 @ B3 )
& ( factor8216151070175719842xt_a_b @ G2 @ B3 @ A4 ) ) ) ) ).
% associated_def
thf(fact_1128_associated__def,axiom,
( associ6879500422977059064t_unit
= ( ^ [G2: partia8223610829204095565t_unit,A4: a,B3: a] :
( ( factor3040189038382604065t_unit @ G2 @ A4 @ B3 )
& ( factor3040189038382604065t_unit @ G2 @ B3 @ A4 ) ) ) ) ).
% associated_def
thf(fact_1129_associated__def,axiom,
( associ5603075271488036121t_unit
= ( ^ [G2: partia2956882679547061052t_unit,A4: list_list_a,B3: list_list_a] :
( ( factor6954119973539764400t_unit @ G2 @ A4 @ B3 )
& ( factor6954119973539764400t_unit @ G2 @ B3 @ A4 ) ) ) ) ).
% associated_def
thf(fact_1130_associatedE,axiom,
! [G: partia2670972154091845814t_unit,A2: list_a,B: list_a] :
( ( associ8407585678920448409t_unit @ G @ A2 @ B )
=> ~ ( ( factor1757716651909850160t_unit @ G @ A2 @ B )
=> ~ ( factor1757716651909850160t_unit @ G @ B @ A2 ) ) ) ).
% associatedE
thf(fact_1131_associatedE,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B: a] :
( ( associ5860276527279195403xt_a_b @ G @ A2 @ B )
=> ~ ( ( factor8216151070175719842xt_a_b @ G @ A2 @ B )
=> ~ ( factor8216151070175719842xt_a_b @ G @ B @ A2 ) ) ) ).
% associatedE
thf(fact_1132_associatedE,axiom,
! [G: partia8223610829204095565t_unit,A2: a,B: a] :
( ( associ6879500422977059064t_unit @ G @ A2 @ B )
=> ~ ( ( factor3040189038382604065t_unit @ G @ A2 @ B )
=> ~ ( factor3040189038382604065t_unit @ G @ B @ A2 ) ) ) ).
% associatedE
thf(fact_1133_associatedE,axiom,
! [G: partia2956882679547061052t_unit,A2: list_list_a,B: list_list_a] :
( ( associ5603075271488036121t_unit @ G @ A2 @ B )
=> ~ ( ( factor6954119973539764400t_unit @ G @ A2 @ B )
=> ~ ( factor6954119973539764400t_unit @ G @ B @ A2 ) ) ) ).
% associatedE
thf(fact_1134_associatedD,axiom,
! [G: partia2670972154091845814t_unit,A2: list_a,B: list_a] :
( ( associ8407585678920448409t_unit @ G @ A2 @ B )
=> ( factor1757716651909850160t_unit @ G @ A2 @ B ) ) ).
% associatedD
thf(fact_1135_associatedD,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B: a] :
( ( associ5860276527279195403xt_a_b @ G @ A2 @ B )
=> ( factor8216151070175719842xt_a_b @ G @ A2 @ B ) ) ).
% associatedD
thf(fact_1136_associatedD,axiom,
! [G: partia8223610829204095565t_unit,A2: a,B: a] :
( ( associ6879500422977059064t_unit @ G @ A2 @ B )
=> ( factor3040189038382604065t_unit @ G @ A2 @ B ) ) ).
% associatedD
thf(fact_1137_associatedD,axiom,
! [G: partia2956882679547061052t_unit,A2: list_list_a,B: list_list_a] :
( ( associ5603075271488036121t_unit @ G @ A2 @ B )
=> ( factor6954119973539764400t_unit @ G @ A2 @ B ) ) ).
% associatedD
thf(fact_1138_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B: a,B5: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B5 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A2 @ B ) @ ( mult_a_ring_ext_a_b @ G @ A2 @ B5 ) )
=> ( associ5860276527279195403xt_a_b @ G @ B @ B5 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_1139_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia2670972154091845814t_unit,A2: list_a,B: list_a,B5: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B5 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( associ8407585678920448409t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ A2 @ B ) @ ( mult_l7073676228092353617t_unit @ G @ A2 @ B5 ) )
=> ( associ8407585678920448409t_unit @ G @ B @ B5 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_1140_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia2956882679547061052t_unit,A2: list_list_a,B: list_list_a,B5: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( member_list_list_a @ A2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ B5 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( associ5603075271488036121t_unit @ G @ ( mult_l4853965630390486993t_unit @ G @ A2 @ B ) @ ( mult_l4853965630390486993t_unit @ G @ A2 @ B5 ) )
=> ( associ5603075271488036121t_unit @ G @ B @ B5 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_1141_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia8223610829204095565t_unit,A2: a,B: a,B5: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B5 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( associ6879500422977059064t_unit @ G @ ( mult_a_Product_unit @ G @ A2 @ B ) @ ( mult_a_Product_unit @ G @ A2 @ B5 ) )
=> ( associ6879500422977059064t_unit @ G @ B @ B5 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_1142_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A2 @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ A2 @ B )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_1143_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia2670972154091845814t_unit,A2: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( associ8407585678920448409t_unit @ G @ A2 @ B )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_1144_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia2956882679547061052t_unit,A2: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( member_list_list_a @ A2 @ ( units_4903515905731149798t_unit @ G ) )
=> ( ( associ5603075271488036121t_unit @ G @ A2 @ B )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
=> ( member_list_list_a @ B @ ( units_4903515905731149798t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_1145_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia8223610829204095565t_unit,A2: a,B: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( member_a @ A2 @ ( units_a_Product_unit @ G ) )
=> ( ( associ6879500422977059064t_unit @ G @ A2 @ B )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ B @ ( units_a_Product_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_1146_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A2 @ B )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ A2 @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_1147_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia2670972154091845814t_unit,A2: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A2 @ B )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_1148_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia2956882679547061052t_unit,A2: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( associ5603075271488036121t_unit @ G @ A2 @ B )
=> ( ( member_list_list_a @ B @ ( units_4903515905731149798t_unit @ G ) )
=> ( ( member_list_list_a @ A2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( member_list_list_a @ A2 @ ( units_4903515905731149798t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_1149_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia8223610829204095565t_unit,A2: a,B: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A2 @ B )
=> ( ( member_a @ B @ ( units_a_Product_unit @ G ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( member_a @ A2 @ ( units_a_Product_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_1150_monoid__cancel_Oirreducible__cong,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,A5: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( irredu6211895646901577903xt_a_b @ G @ A2 )
=> ( ( associ5860276527279195403xt_a_b @ G @ A2 @ A5 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( irredu6211895646901577903xt_a_b @ G @ A5 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_1151_monoid__cancel_Oirreducible__cong,axiom,
! [G: partia2670972154091845814t_unit,A2: list_a,A5: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( irredu4230924414530676029t_unit @ G @ A2 )
=> ( ( associ8407585678920448409t_unit @ G @ A2 @ A5 )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ G ) )
=> ( irredu4230924414530676029t_unit @ G @ A5 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_1152_monoid__cancel_Oirreducible__cong,axiom,
! [G: partia2956882679547061052t_unit,A2: list_list_a,A5: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( irredu4439051761327310013t_unit @ G @ A2 )
=> ( ( associ5603075271488036121t_unit @ G @ A2 @ A5 )
=> ( ( member_list_list_a @ A2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ A5 @ ( partia2464479390973590831t_unit @ G ) )
=> ( irredu4439051761327310013t_unit @ G @ A5 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_1153_monoid__cancel_Oirreducible__cong,axiom,
! [G: partia8223610829204095565t_unit,A2: a,A5: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( irredu4023057619401689684t_unit @ G @ A2 )
=> ( ( associ6879500422977059064t_unit @ G @ A2 @ A5 )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ A5 @ ( partia6735698275553448452t_unit @ G ) )
=> ( irredu4023057619401689684t_unit @ G @ A5 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_1154_monoid__cancel_Oprime__cong,axiom,
! [G: partia2175431115845679010xt_a_b,P: a,P5: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( prime_a_ring_ext_a_b @ G @ P )
=> ( ( associ5860276527279195403xt_a_b @ G @ P @ P5 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ P5 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( prime_a_ring_ext_a_b @ G @ P5 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_1155_monoid__cancel_Oprime__cong,axiom,
! [G: partia2670972154091845814t_unit,P: list_a,P5: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( prime_2011924034616061926t_unit @ G @ P )
=> ( ( associ8407585678920448409t_unit @ G @ P @ P5 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ P5 @ ( partia5361259788508890537t_unit @ G ) )
=> ( prime_2011924034616061926t_unit @ G @ P5 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_1156_monoid__cancel_Oprime__cong,axiom,
! [G: partia2956882679547061052t_unit,P: list_list_a,P5: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( prime_1232919612140715622t_unit @ G @ P )
=> ( ( associ5603075271488036121t_unit @ G @ P @ P5 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ P5 @ ( partia2464479390973590831t_unit @ G ) )
=> ( prime_1232919612140715622t_unit @ G @ P5 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_1157_monoid__cancel_Oprime__cong,axiom,
! [G: partia8223610829204095565t_unit,P: a,P5: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( prime_a_Product_unit @ G @ P )
=> ( ( associ6879500422977059064t_unit @ G @ P @ P5 )
=> ( ( member_a @ P @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ P5 @ ( partia6735698275553448452t_unit @ G ) )
=> ( prime_a_Product_unit @ G @ P5 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_1158_divides__irreducible__condition,axiom,
! [G: partia2175431115845679010xt_a_b,R3: a,A2: a] :
( ( irredu6211895646901577903xt_a_b @ G @ R3 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( factor8216151070175719842xt_a_b @ G @ A2 @ R3 )
=> ( ( member_a @ A2 @ ( units_a_ring_ext_a_b @ G ) )
| ( associ5860276527279195403xt_a_b @ G @ A2 @ R3 ) ) ) ) ) ).
% divides_irreducible_condition
thf(fact_1159_divides__irreducible__condition,axiom,
! [G: partia2670972154091845814t_unit,R3: list_a,A2: list_a] :
( ( irredu4230924414530676029t_unit @ G @ R3 )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( factor1757716651909850160t_unit @ G @ A2 @ R3 )
=> ( ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ G ) )
| ( associ8407585678920448409t_unit @ G @ A2 @ R3 ) ) ) ) ) ).
% divides_irreducible_condition
thf(fact_1160_divides__irreducible__condition,axiom,
! [G: partia2956882679547061052t_unit,R3: list_list_a,A2: list_list_a] :
( ( irredu4439051761327310013t_unit @ G @ R3 )
=> ( ( member_list_list_a @ A2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( factor6954119973539764400t_unit @ G @ A2 @ R3 )
=> ( ( member_list_list_a @ A2 @ ( units_4903515905731149798t_unit @ G ) )
| ( associ5603075271488036121t_unit @ G @ A2 @ R3 ) ) ) ) ) ).
% divides_irreducible_condition
thf(fact_1161_divides__irreducible__condition,axiom,
! [G: partia8223610829204095565t_unit,R3: a,A2: a] :
( ( irredu4023057619401689684t_unit @ G @ R3 )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( factor3040189038382604065t_unit @ G @ A2 @ R3 )
=> ( ( member_a @ A2 @ ( units_a_Product_unit @ G ) )
| ( associ6879500422977059064t_unit @ G @ A2 @ R3 ) ) ) ) ) ).
% divides_irreducible_condition
thf(fact_1162_monoid__cancel_Oassociated__iff,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ A2 @ B )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ G ) )
& ( A2
= ( mult_a_ring_ext_a_b @ G @ B @ X2 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_1163_monoid__cancel_Oassociated__iff,axiom,
! [G: partia2670972154091845814t_unit,A2: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( associ8407585678920448409t_unit @ G @ A2 @ B )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ G ) )
& ( A2
= ( mult_l7073676228092353617t_unit @ G @ B @ X2 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_1164_monoid__cancel_Oassociated__iff,axiom,
! [G: partia2956882679547061052t_unit,A2: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( member_list_list_a @ A2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( associ5603075271488036121t_unit @ G @ A2 @ B )
= ( ? [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( units_4903515905731149798t_unit @ G ) )
& ( A2
= ( mult_l4853965630390486993t_unit @ G @ B @ X2 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_1165_monoid__cancel_Oassociated__iff,axiom,
! [G: partia8223610829204095565t_unit,A2: a,B: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( associ6879500422977059064t_unit @ G @ A2 @ B )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_Product_unit @ G ) )
& ( A2
= ( mult_a_Product_unit @ G @ B @ X2 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_1166_monoid__cancel_OassociatedE2,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A2 @ B )
=> ( ! [U2: a] :
( ( A2
= ( mult_a_ring_ext_a_b @ G @ B @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_ring_ext_a_b @ G ) ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ~ ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_1167_monoid__cancel_OassociatedE2,axiom,
! [G: partia2670972154091845814t_unit,A2: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A2 @ B )
=> ( ! [U2: list_a] :
( ( A2
= ( mult_l7073676228092353617t_unit @ G @ B @ U2 ) )
=> ~ ( member_list_a @ U2 @ ( units_2932844235741507942t_unit @ G ) ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
=> ~ ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_1168_monoid__cancel_OassociatedE2,axiom,
! [G: partia2956882679547061052t_unit,A2: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( associ5603075271488036121t_unit @ G @ A2 @ B )
=> ( ! [U2: list_list_a] :
( ( A2
= ( mult_l4853965630390486993t_unit @ G @ B @ U2 ) )
=> ~ ( member_list_list_a @ U2 @ ( units_4903515905731149798t_unit @ G ) ) )
=> ( ( member_list_list_a @ A2 @ ( partia2464479390973590831t_unit @ G ) )
=> ~ ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_1169_monoid__cancel_OassociatedE2,axiom,
! [G: partia8223610829204095565t_unit,A2: a,B: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A2 @ B )
=> ( ! [U2: a] :
( ( A2
= ( mult_a_Product_unit @ G @ B @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_Product_unit @ G ) ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
=> ~ ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_1170_monoid__cancel_OassociatedD2,axiom,
! [G: partia2175431115845679010xt_a_b,A2: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A2 @ B )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
& ( A2
= ( mult_a_ring_ext_a_b @ G @ B @ X ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_1171_monoid__cancel_OassociatedD2,axiom,
! [G: partia2670972154091845814t_unit,A2: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A2 @ B )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ? [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
& ( A2
= ( mult_l7073676228092353617t_unit @ G @ B @ X ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_1172_monoid__cancel_OassociatedD2,axiom,
! [G: partia2956882679547061052t_unit,A2: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( associ5603075271488036121t_unit @ G @ A2 @ B )
=> ( ( member_list_list_a @ A2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
=> ? [X: list_list_a] :
( ( member_list_list_a @ X @ ( units_4903515905731149798t_unit @ G ) )
& ( A2
= ( mult_l4853965630390486993t_unit @ G @ B @ X ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_1173_monoid__cancel_OassociatedD2,axiom,
! [G: partia8223610829204095565t_unit,A2: a,B: a] :
( ( monoid1999574367301118026t_unit @ G )
=> ( ( associ6879500422977059064t_unit @ G @ A2 @ B )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ G ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ G ) )
=> ? [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ G ) )
& ( A2
= ( mult_a_Product_unit @ G @ B @ X ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_1174_x_Oindependent__rotate1,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( append_list_a @ Us2 @ Vs ) )
=> ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( append_list_a @ ( rotate1_list_a @ Us2 ) @ Vs ) ) ) ) ).
% x.independent_rotate1
thf(fact_1175_x_Otrivial__combine__imp__independent,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [Ks: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks ) @ K )
=> ( ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ Us2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( take_list_a @ ( size_s349497388124573686list_a @ Us2 ) @ Ks ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) )
=> ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ).
% x.trivial_combine_imp_independent
thf(fact_1176_associated__sym,axiom,
! [A2: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A2 @ B )
=> ( associ5860276527279195403xt_a_b @ r @ B @ A2 ) ) ).
% associated_sym
thf(fact_1177_associated__trans,axiom,
! [A2: a,B: a,C: a] :
( ( associ5860276527279195403xt_a_b @ r @ A2 @ B )
=> ( ( associ5860276527279195403xt_a_b @ r @ B @ C )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A2 @ C ) ) ) ) ) ).
% associated_trans
thf(fact_1178_assoc__subst,axiom,
! [A2: a,B: a,F: a > a] :
( ( associ5860276527279195403xt_a_b @ r @ A2 @ B )
=> ( ! [A3: a,B2: a] :
( ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ A3 @ B2 ) )
=> ( ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ ( F @ B2 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ ( F @ A3 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( F @ A2 ) @ ( F @ B ) ) ) ) ) ) ).
% assoc_subst
thf(fact_1179_mult__of_Oassociated__sym,axiom,
! [A2: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ A2 ) ) ).
% mult_of.associated_sym
thf(fact_1180_Units__assoc,axiom,
! [A2: a,B: a] :
( ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A2 @ B ) ) ) ).
% Units_assoc
thf(fact_1181_assoc__iff__assoc__mult,axiom,
! [A2: a,B: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A2 @ B )
= ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B ) ) ) ) ).
% assoc_iff_assoc_mult
thf(fact_1182_mult__cong__r,axiom,
! [B: a,B5: a,A2: a] :
( ( associ5860276527279195403xt_a_b @ r @ B @ B5 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B5 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_1183_mult__cong__l,axiom,
! [A2: a,A5: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A2 @ A5 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) @ ( mult_a_ring_ext_a_b @ r @ A5 @ B ) ) ) ) ) ) ).
% mult_cong_l
thf(fact_1184_mult__of_Oassoc__subst,axiom,
! [A2: a,B: a,F: a > a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( ! [A3: a,B2: a] :
( ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A3 @ B2 ) )
=> ( ( member_a @ ( F @ A3 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( member_a @ ( F @ B2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( F @ A3 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( F @ A2 ) @ ( F @ B ) ) ) ) ) ) ).
% mult_of.assoc_subst
thf(fact_1185_mult__of_Oassociated__trans,axiom,
! [A2: a,B: a,C: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ C ) ) ) ) ) ).
% mult_of.associated_trans
thf(fact_1186_Units__cong,axiom,
! [A2: a,B: a] :
( ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A2 @ B )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_cong
thf(fact_1187_divides__cong__r,axiom,
! [X3: a,Y: a,Y2: a] :
( ( factor8216151070175719842xt_a_b @ r @ X3 @ Y )
=> ( ( associ5860276527279195403xt_a_b @ r @ Y @ Y2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ X3 @ Y2 ) ) ) ) ).
% divides_cong_r
thf(fact_1188_divides__cong__l,axiom,
! [X3: a,X5: a,Y: a] :
( ( associ5860276527279195403xt_a_b @ r @ X3 @ X5 )
=> ( ( factor8216151070175719842xt_a_b @ r @ X5 @ Y )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ X3 @ Y ) ) ) ) ).
% divides_cong_l
thf(fact_1189_mult__of_OUnits__assoc,axiom,
! [A2: a,B: a] :
( ( member_a @ A2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B ) ) ) ).
% mult_of.Units_assoc
thf(fact_1190_mult__of_Oassoc__l__cancel,axiom,
! [A2: a,B: a,B5: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B5 ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ B5 ) ) ) ) ) ).
% mult_of.assoc_l_cancel
thf(fact_1191_mult__of_Oassoc__r__cancel,axiom,
! [A2: a,B: a,A5: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) @ ( mult_a_ring_ext_a_b @ r @ A5 @ B ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ A5 ) ) ) ) ) ).
% mult_of.assoc_r_cancel
thf(fact_1192_mult__of_Omult__cong__l,axiom,
! [A2: a,A5: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ A5 )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) @ ( mult_a_ring_ext_a_b @ r @ A5 @ B ) ) ) ) ) ) ).
% mult_of.mult_cong_l
thf(fact_1193_mult__of_Omult__cong__r,axiom,
! [B: a,B5: a,A2: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ B5 )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B5 ) ) ) ) ) ) ).
% mult_of.mult_cong_r
thf(fact_1194_associatedI2,axiom,
! [U: a,A2: a,B: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( A2
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A2 @ B ) ) ) ) ).
% associatedI2
thf(fact_1195_associatedI2_H,axiom,
! [A2: a,B: a,U: a] :
( ( A2
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A2 @ B ) ) ) ) ).
% associatedI2'
thf(fact_1196_ring__associated__iff,axiom,
! [A2: a,B: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A2 @ B )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
& ( A2
= ( mult_a_ring_ext_a_b @ r @ X2 @ B ) ) ) ) ) ) ) ).
% ring_associated_iff
thf(fact_1197_mult__of_Odivides__cong__l,axiom,
! [X3: a,X5: a,Y: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ X3 @ X5 )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X5 @ Y )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X3 @ Y ) ) ) ) ).
% mult_of.divides_cong_l
thf(fact_1198_mult__of_Odivides__cong__r,axiom,
! [X3: a,Y: a,Y2: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X3 @ Y )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ Y @ Y2 )
=> ( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X3 @ Y2 ) ) ) ) ).
% mult_of.divides_cong_r
thf(fact_1199_mult__of_Oassoc__unit__r,axiom,
! [A2: a,B: a] :
( ( member_a @ A2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.assoc_unit_r
thf(fact_1200_mult__of_Oassoc__unit__l,axiom,
! [A2: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.assoc_unit_l
thf(fact_1201_mult__of_Oirreducible__cong,axiom,
! [A2: a,A5: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A2 )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ A5 )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A5 ) ) ) ) ) ).
% mult_of.irreducible_cong
thf(fact_1202_mult__of_Oprime__cong,axiom,
! [P: a,P5: a] :
( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ P @ P5 )
=> ( ( member_a @ P @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ P5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P5 ) ) ) ) ) ).
% mult_of.prime_cong
thf(fact_1203_mult__of_Oassociated__fcount,axiom,
! [A2: a,B: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ A2 )
= ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ B ) ) ) ) ) ).
% mult_of.associated_fcount
thf(fact_1204_mult__of_Ogcdof__cong__l,axiom,
! [A5: a,A2: a,B: a,C: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A5 @ A2 )
=> ( ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B @ C )
=> ( ( member_a @ A5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A5 @ B @ C ) ) ) ) ) ) ) ).
% mult_of.gcdof_cong_l
thf(fact_1205_mult__of_OassociatedD2,axiom,
! [A2: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( A2
= ( mult_a_ring_ext_a_b @ r @ B @ X ) ) ) ) ) ) ).
% mult_of.associatedD2
thf(fact_1206_mult__of_OassociatedE2,axiom,
! [A2: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( ! [U2: a] :
( ( A2
= ( mult_a_ring_ext_a_b @ r @ B @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ~ ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.associatedE2
thf(fact_1207_mult__of_OassociatedI2,axiom,
! [U: a,A2: a,B: a] :
( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( A2
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B ) ) ) ) ).
% mult_of.associatedI2
thf(fact_1208_mult__of_OassociatedI2_H,axiom,
! [A2: a,B: a,U: a] :
( ( A2
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B ) ) ) ) ).
% mult_of.associatedI2'
thf(fact_1209_mult__of_Oassociated__iff,axiom,
! [A2: a,B: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( A2
= ( mult_a_ring_ext_a_b @ r @ B @ X2 ) ) ) ) ) ) ) ).
% mult_of.associated_iff
thf(fact_1210_x_Odivides__pirreducible__condition,axiom,
! [K: set_list_a,Q: list_list_a,P: list_list_a] :
( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ Q )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ Q )
=> ( ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
| ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ Q ) ) ) ) ) ).
% x.divides_pirreducible_condition
thf(fact_1211_x_Ocombine__take,axiom,
! [Us2: list_list_a,Ks2: list_list_a] :
( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( take_list_a @ ( size_s349497388124573686list_a @ Us2 ) @ Ks2 ) @ Us2 )
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ Us2 ) ) ).
% x.combine_take
thf(fact_1212_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_1213_rotate1__is__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( ( rotate1_list_a @ Xs )
= nil_list_a )
= ( Xs = nil_list_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_1214_associated__refl,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A2 @ A2 ) ) ).
% associated_refl
thf(fact_1215_take__all__iff,axiom,
! [N: nat,Xs: list_list_a] :
( ( ( take_list_a @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_1216_take__all__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ( take_a @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_1217_take__all,axiom,
! [Xs: list_list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N )
=> ( ( take_list_a @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_1218_take__all,axiom,
! [Xs: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( take_a @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_1219_mult__of_Oassociated__refl,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ A2 ) ) ).
% mult_of.associated_refl
thf(fact_1220_x_Onon__trivial__combine__imp__dependent,axiom,
! [K: set_list_a,Ks2: list_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks2 ) @ K )
=> ( ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ Us2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ~ ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( take_list_a @ ( size_s349497388124573686list_a @ Us2 ) @ Ks2 ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
=> ~ ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ) ).
% x.non_trivial_combine_imp_dependent
thf(fact_1221_x_Oindependent__imp__trivial__combine,axiom,
! [K: set_list_a,Us2: list_list_a,Ks2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks2 ) @ K )
=> ( ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ Us2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( take_list_a @ ( size_s349497388124573686list_a @ Us2 ) @ Ks2 ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ) ) ).
% x.independent_imp_trivial_combine
thf(fact_1222_take__Nil,axiom,
! [N: nat] :
( ( take_a @ N @ nil_a )
= nil_a ) ).
% take_Nil
thf(fact_1223_take__Nil,axiom,
! [N: nat] :
( ( take_list_a @ N @ nil_list_a )
= nil_list_a ) ).
% take_Nil
thf(fact_1224_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_1225_rotate1_Osimps_I1_J,axiom,
( ( rotate1_list_a @ nil_list_a )
= nil_list_a ) ).
% rotate1.simps(1)
thf(fact_1226_set__take__subset,axiom,
! [N: nat,Xs: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ N @ Xs ) ) @ ( set_a2 @ Xs ) ) ).
% set_take_subset
thf(fact_1227_set__take__subset,axiom,
! [N: nat,Xs: list_list_a] : ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( take_list_a @ N @ Xs ) ) @ ( set_list_a2 @ Xs ) ) ).
% set_take_subset
thf(fact_1228_set__take__subset__set__take,axiom,
! [M2: nat,N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ M2 @ Xs ) ) @ ( set_a2 @ ( take_a @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1229_set__take__subset__set__take,axiom,
! [M2: nat,N: nat,Xs: list_list_a] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( take_list_a @ M2 @ Xs ) ) @ ( set_list_a2 @ ( take_list_a @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1230_rotate1_Osimps_I2_J,axiom,
! [X3: list_a,Xs: list_list_a] :
( ( rotate1_list_a @ ( cons_list_a @ X3 @ Xs ) )
= ( append_list_a @ Xs @ ( cons_list_a @ X3 @ nil_list_a ) ) ) ).
% rotate1.simps(2)
thf(fact_1231_rotate1_Osimps_I2_J,axiom,
! [X3: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X3 @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X3 @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_1232_x_Ocombine__normalize,axiom,
! [Ks2: list_list_a,Us2: list_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ Us2 )
= A2 )
=> ~ ! [Ks5: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( take_list_a @ ( size_s349497388124573686list_a @ Us2 ) @ Ks2 ) ) @ ( set_list_a2 @ Ks5 ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks5 ) @ ( sup_sup_set_list_a @ ( set_list_a2 @ ( take_list_a @ ( size_s349497388124573686list_a @ Us2 ) @ Ks2 ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( ( size_s349497388124573686list_a @ Ks5 )
= ( size_s349497388124573686list_a @ Us2 ) )
=> ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks5 @ Us2 )
!= A2 ) ) ) ) ) ) ) ).
% x.combine_normalize
thf(fact_1233_x_Oindependent__append,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs )
=> ( ( ( inf_inf_set_list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
=> ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( append_list_a @ Us2 @ Vs ) ) ) ) ) ) ).
% x.independent_append
thf(fact_1234_x_Osubring__inter,axiom,
! [I: set_list_a,J: set_list_a] :
( ( subrin6918843898125473962t_unit @ I @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subrin6918843898125473962t_unit @ J @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subrin6918843898125473962t_unit @ ( inf_inf_set_list_a @ I @ J ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subring_inter
thf(fact_1235_x_Osubalgebra__inter,axiom,
! [K: set_list_a,V: set_list_a,V5: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1768981623711841426t_unit @ K @ V5 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( embedd1768981623711841426t_unit @ K @ ( inf_inf_set_list_a @ V @ V5 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subalgebra_inter
thf(fact_1236_x_Osubcring__inter,axiom,
! [I: set_list_a,J: set_list_a] :
( ( subcri7763218559781929323t_unit @ I @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subcri7763218559781929323t_unit @ J @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subcri7763218559781929323t_unit @ ( inf_inf_set_list_a @ I @ J ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subcring_inter
thf(fact_1237_x_Oindependent__split_I3_J,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( append_list_a @ Us2 @ Vs ) )
=> ( ( inf_inf_set_list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ).
% x.independent_split(3)
thf(fact_1238_mult__of_Omultlist__ee__cong,axiom,
! [Fs: list_a,Fs4: list_a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ Fs4 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs4 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs @ ( one_a_ring_ext_a_b @ r ) ) @ ( foldr_a_a @ ( mult_a_ring_ext_a_b @ r ) @ Fs4 @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% mult_of.multlist_ee_cong
thf(fact_1239_mult__of_Oee__factorsI,axiom,
! [A2: a,B: a,As: list_a,Bs: list_a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A2 )
=> ( ~ ( member_a @ A2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B )
=> ( ~ ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs ) ) ) ) ) ) ) ) ).
% mult_of.ee_factorsI
thf(fact_1240_mult__of_Oee__length,axiom,
! [As: list_a,Bs: list_a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) ) ) ).
% mult_of.ee_length
thf(fact_1241_mult__of_Oee__sym,axiom,
! [As: list_a,Bs: list_a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ As ) ) ) ) ).
% mult_of.ee_sym
thf(fact_1242_mult__of_Oee__trans,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ Cs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Cs ) ) ) ) ) ) ).
% mult_of.ee_trans
thf(fact_1243_mult__of_Oee__factorsD,axiom,
! [As: list_a,Bs: list_a,A2: a,B: a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A2 )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B ) ) ) ) ) ) ).
% mult_of.ee_factorsD
thf(fact_1244_mult__of_Ofactors__unique,axiom,
! [Fs: list_a,A2: a,Fs4: list_a] :
( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A2 )
=> ( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ Fs4 @ A2 )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ~ ( member_a @ A2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs4 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ Fs4 ) ) ) ) ) ) ) ).
% mult_of.factors_unique
thf(fact_1245_mult__of_Oee__refl,axiom,
! [As: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ As ) ) ).
% mult_of.ee_refl
thf(fact_1246_finite__Diff__insert,axiom,
! [A: set_list_a,A2: list_a,B4: set_list_a] :
( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ ( insert_list_a @ A2 @ B4 ) ) )
= ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_1247_finite__Diff__insert,axiom,
! [A: set_a,A2: a,B4: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B4 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_1248_Diff__eq__empty__iff,axiom,
! [A: set_a,B4: set_a] :
( ( ( minus_minus_set_a @ A @ B4 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_1249_Diff__eq__empty__iff,axiom,
! [A: set_list_a,B4: set_list_a] :
( ( ( minus_646659088055828811list_a @ A @ B4 )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_1250_subset__antisym,axiom,
! [A: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A )
=> ( A = B4 ) ) ) ).
% subset_antisym
thf(fact_1251_subset__antisym,axiom,
! [A: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B4 )
=> ( ( ord_le8861187494160871172list_a @ B4 @ A )
=> ( A = B4 ) ) ) ).
% subset_antisym
thf(fact_1252_subsetI,axiom,
! [A: set_list_list_a,B4: set_list_list_a] :
( ! [X: list_list_a] :
( ( member_list_list_a @ X @ A )
=> ( member_list_list_a @ X @ B4 ) )
=> ( ord_le8488217952732425610list_a @ A @ B4 ) ) ).
% subsetI
thf(fact_1253_subsetI,axiom,
! [A: set_a,B4: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ X @ B4 ) )
=> ( ord_less_eq_set_a @ A @ B4 ) ) ).
% subsetI
thf(fact_1254_subsetI,axiom,
! [A: set_list_a,B4: set_list_a] :
( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( member_list_a @ X @ B4 ) )
=> ( ord_le8861187494160871172list_a @ A @ B4 ) ) ).
% subsetI
thf(fact_1255_factorization__property,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) )
=> ? [Fs3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs3 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs3 @ A2 ) ) ) ) ).
% factorization_property
thf(fact_1256_x_Omaximalideal__prime,axiom,
! [I: set_list_a] :
( ( maxima6585700282301356660t_unit @ I @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( primei6309817859076077608t_unit @ I @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.maximalideal_prime
thf(fact_1257_subring__inter,axiom,
! [I: set_a,J: set_a] :
( ( subring_a_b @ I @ r )
=> ( ( subring_a_b @ J @ r )
=> ( subring_a_b @ ( inf_inf_set_a @ I @ J ) @ r ) ) ) ).
% subring_inter
thf(fact_1258_subalgebra__inter,axiom,
! [K: set_a,V: set_a,V5: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V5 @ r )
=> ( embedd9027525575939734154ra_a_b @ K @ ( inf_inf_set_a @ V @ V5 ) @ r ) ) ) ).
% subalgebra_inter
thf(fact_1259_subcring__inter,axiom,
! [I: set_a,J: set_a] :
( ( subcring_a_b @ I @ r )
=> ( ( subcring_a_b @ J @ r )
=> ( subcring_a_b @ ( inf_inf_set_a @ I @ J ) @ r ) ) ) ).
% subcring_inter
thf(fact_1260_mult__of_Ounit__wfactors,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ nil_a @ A2 ) ) ).
% mult_of.unit_wfactors
thf(fact_1261_mult__of_Ofactorcount__exists,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [C2: nat] :
! [As2: list_a] :
( ( ( ord_less_eq_set_a @ ( set_a2 @ As2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As2 @ A2 ) )
=> ( C2
= ( size_size_list_a @ As2 ) ) ) ) ).
% mult_of.factorcount_exists
thf(fact_1262_mult__of_Owfactors__cong__r,axiom,
! [Fs: list_a,A2: a,A5: a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A2 )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ A5 )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A5 ) ) ) ) ) ) ).
% mult_of.wfactors_cong_r
thf(fact_1263_mult__of_Owfactors__dividesI,axiom,
! [Fs: list_a,A2: a,F: a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ F @ A2 ) ) ) ) ) ).
% mult_of.wfactors_dividesI
thf(fact_1264_mult__of_Owfactors__perm__cong__l,axiom,
! [Fs: list_a,A2: a,Fs4: list_a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A2 )
=> ( ( ( mset_a @ Fs )
= ( mset_a @ Fs4 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs4 @ A2 ) ) ) ) ).
% mult_of.wfactors_perm_cong_l
thf(fact_1265_mult__of_Owfactors__unique,axiom,
! [Fs: list_a,A2: a,Fs4: list_a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A2 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs4 @ A2 )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs4 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ Fs4 ) ) ) ) ) ) ).
% mult_of.wfactors_unique
thf(fact_1266_mult__of_Owfactors__ee__cong__l,axiom,
! [As: list_a,Bs: list_a,B: a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ B ) ) ) ) ) ) ).
% mult_of.wfactors_ee_cong_l
thf(fact_1267_mult__of_Ofactors__wfactors,axiom,
! [As: list_a,A2: a] :
( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A2 ) ) ) ).
% mult_of.factors_wfactors
thf(fact_1268_mult__of_Ounit__wfactors__empty,axiom,
! [A2: a,Fs: list_a] :
( ( member_a @ A2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( Fs = nil_a ) ) ) ) ).
% mult_of.unit_wfactors_empty
thf(fact_1269_mult__of_Operm__wfactorsD,axiom,
! [As: list_a,Bs: list_a,A2: a,B: a] :
( ( ( mset_a @ As )
= ( mset_a @ Bs ) )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A2 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B ) ) ) ) ) ) ) ).
% mult_of.perm_wfactorsD
thf(fact_1270_mult__of_Oee__wfactorsI,axiom,
! [A2: a,B: a,As: list_a,Bs: list_a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A2 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs ) ) ) ) ) ) ) ) ).
% mult_of.ee_wfactorsI
thf(fact_1271_mult__of_Oee__wfactorsD,axiom,
! [As: list_a,Bs: list_a,A2: a,B: a] :
( ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A2 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B ) ) ) ) ) ) ) ) ).
% mult_of.ee_wfactorsD
thf(fact_1272_mult__of_Oee__wfactors,axiom,
! [As: list_a,A2: a,Bs: list_a,B: a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A2 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A2 @ B )
= ( essent9005414202370111435t_unit @ ( ring_mult_of_a_b @ r ) @ As @ Bs ) ) ) ) ) ) ) ) ).
% mult_of.ee_wfactors
thf(fact_1273_mult__of_Owfactors__factors,axiom,
! [As: list_a,A2: a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [A6: a] :
( ( factor4979495158039764464t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A6 )
& ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A6 @ A2 ) ) ) ) ).
% mult_of.wfactors_factors
thf(fact_1274_mult__of_Ofactorcount__unique,axiom,
! [As: list_a,A2: a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A2 )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ A2 )
= ( size_size_list_a @ As ) ) ) ) ) ).
% mult_of.factorcount_unique
thf(fact_1275_mult__of_Owfactors__mult__single,axiom,
! [A2: a,Fb: list_a,B: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A2 )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fb @ B )
=> ( ( member_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ ( cons_a @ A2 @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A2 @ B ) ) ) ) ) ) ) ).
% mult_of.wfactors_mult_single
% Conjectures (1)
thf(conj_0,conjecture,
( ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ s2 ) @ s )
= ( zero_a_b @ r ) ) ).
%------------------------------------------------------------------------------