TPTP Problem File: SLH0267^1.p
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%------------------------------------------------------------------------------
% 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_00221_008455__17330696_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1634 ( 426 unt; 350 typ; 0 def)
% Number of atoms : 4119 (1147 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 17180 ( 269 ~; 53 |; 213 &;14503 @)
% ( 0 <=>;2142 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 32 ( 31 usr)
% Number of type conns : 1333 (1333 >; 0 *; 0 +; 0 <<)
% Number of symbols : 322 ( 319 usr; 16 con; 0-4 aty)
% Number of variables : 3346 ( 481 ^;2758 !; 107 ?;3346 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:38:07.334
%------------------------------------------------------------------------------
% Could-be-implicit typings (31)
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% Explicit typings (319)
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thf(sy_c_Ring__Divisibility_Oeuclidean__domain_001tf__a_001tf__b,type,
ring_e8745995371659049232in_a_b: partia2175431115845679010xt_a_b > ( a > nat ) > $o ).
thf(sy_c_Ring__Divisibility_Ofactorial__domain_001tf__a_001tf__b,type,
ring_f5272581269873410839in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_n4705423059119889713t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__domain_001tf__a_001tf__b,type,
ring_n4045954140777738665in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_n1719824158142654231t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_n5188127996776581661t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_n7704429503468267069t_unit: partia7496981018696276118t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001tf__a_001tf__b,type,
ring_n3639167112692572309ng_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_p715737262848045090t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_p8098905331641078952t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_p2468016639901664456t_unit: partia7496981018696276118t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001tf__a_001tf__b,type,
ring_p8803135361686045600in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r360171070648044744t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r932985474545269838t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r5115406448772830318t_unit: partia7496981018696276118t_unit > set_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001tf__a_001tf__b,type,
ring_r999134135267193926le_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r5437400583859147359t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r6430282645014804837t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r1091214237498979717t_unit: partia7496981018696276118t_unit > set_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001tf__a_001tf__b,type,
ring_ring_prime_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Set_OCollect_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
collect_list_a_a: ( ( list_a > a ) > $o ) > set_list_a_a ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
collect_nat_list_a: ( ( nat > list_a ) > $o ) > set_nat_list_a ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
collect_set_list_a_a: ( ( set_list_a > a ) > $o ) > set_set_list_a_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
collect_set_list_a: ( set_list_a > $o ) > set_set_list_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
image_8260866953997875467list_a: ( list_a > list_list_a ) > set_list_a > set_list_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
image_list_a_list_a: ( list_a > list_a ) > set_list_a > set_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001tf__a,type,
image_list_a_a: ( list_a > a ) > set_list_a > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001tf__a,type,
image_set_list_a_a: ( set_list_a > a ) > set_set_list_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_Itf__a_J,type,
image_a_list_a: ( a > list_a ) > set_a > set_list_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
insert_set_list_a: set_list_a > set_set_list_a > set_set_list_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Subrings_Osubcring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subcri7763218559781929323t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001tf__a_001tf__b,type,
subcring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubdomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subdom7821232466298058046t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001tf__a_001tf__b,type,
subdomain_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subfie4546268998243038636t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subfie1779122896746047282t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_UnivPoly_Obound_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
bound_list_list_a: list_list_a > nat > ( nat > list_list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001t__List__Olist_Itf__a_J,type,
bound_list_a: list_a > nat > ( nat > list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bound_set_list_a: set_list_a > nat > ( nat > set_list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
up_lis8963924889346801084t_unit: partia2956882679547061052t_unit > set_nat_list_list_a ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
up_lis8464167429055313730t_unit: partia2670972154091845814t_unit > set_nat_list_a ).
thf(sy_c_UnivPoly_Oup_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
up_set529185716248919906t_unit: partia7496981018696276118t_unit > set_nat_set_list_a ).
thf(sy_c_UnivPoly_Oup_001tf__a_001tf__b,type,
up_a_b: partia2175431115845679010xt_a_b > set_nat_a ).
thf(sy_c_member_001_062_I_062_It__List__Olist_Itf__a_J_Mtf__a_J_Mtf__a_J,type,
member_list_a_a_a: ( ( list_a > a ) > a ) > set_list_a_a_a > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_Mtf__a_J,type,
member_nat_list_a_a: ( ( nat > list_a ) > a ) > set_nat_list_a_a > $o ).
thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
member_nat_a_a: ( ( nat > a ) > a ) > set_nat_a_a > $o ).
thf(sy_c_member_001_062_I_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J_Mtf__a_J,type,
member969817812316227871_a_a_a: ( ( set_list_a > a ) > a ) > set_set_list_a_a_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
member_list_a_a: ( list_a > a ) > set_list_a_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member8650753269014980122list_a: ( nat > list_list_a ) > set_nat_list_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
member_nat_list_a: ( nat > list_a ) > set_nat_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member491565700723299188list_a: ( nat > set_list_a ) > set_nat_set_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
member_set_list_a_a: ( set_list_a > a ) > set_set_list_a_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mtf__a_J,type,
member_set_a_a: ( set_a > a ) > set_set_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
member_a_list_a: ( a > list_a ) > set_a_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Set__Oset_Itf__a_J_J,type,
member_a_set_a: ( a > set_a ) > set_a_set_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_S,type,
s: set_a ).
thf(sy_v_p____,type,
p: list_a ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1278)
thf(fact_0_assms_I1_J,axiom,
finite_finite_a @ s ).
% assms(1)
thf(fact_1_assms_I2_J,axiom,
ord_less_eq_set_a @ s @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(2)
thf(fact_2_assms_I3_J,axiom,
member_a @ x @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ s ) ).
% assms(3)
thf(fact_3_factorial__domain__axioms,axiom,
ring_f5272581269873410839in_a_b @ r ).
% factorial_domain_axioms
thf(fact_4_local_Ofield__axioms,axiom,
field_a_b @ r ).
% local.field_axioms
thf(fact_5_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_6_p__def,axiom,
( p
= ( lagran9092808442999052491ux_a_b @ r @ s ) ) ).
% p_def
thf(fact_7_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_8_minus__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_9_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_10_up__minus__closed,axiom,
! [P: nat > a,Q: nat > a] :
( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I: nat] : ( a_minus_a_b @ r @ ( P @ I ) @ ( Q @ I ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_minus_closed
thf(fact_11_lagrange__aux__eval,axiom,
! [S: set_a,X: a] :
( ( finite_finite_a @ S )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( lagran9092808442999052491ux_a_b @ r @ S ) @ X )
= ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ X ) @ S ) ) ) ) ) ).
% lagrange_aux_eval
thf(fact_12_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_13_finprod__one__eqI,axiom,
! [A: set_list_a_a,F: ( list_a > a ) > a] :
( ! [X2: list_a > a] :
( ( member_list_a_a @ X2 @ A )
=> ( ( F @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro5422611492341532390st_a_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_14_finprod__one__eqI,axiom,
! [A: set_set_list_a_a,F: ( set_list_a > a ) > a] :
( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A )
=> ( ( F @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro4938371440467910406st_a_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_15_finprod__one__eqI,axiom,
! [A: set_nat_list_a,F: ( nat > list_a ) > a] :
( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( ( F @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro4838020199848830884list_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_16_finprod__one__eqI,axiom,
! [A: set_nat_a,F: ( nat > a ) > a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( ( F @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro5839458686994656414_nat_a @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_17_finprod__one__eqI,axiom,
! [A: set_a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( F @ X2 )
= ( 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_18_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_19_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_20_eval__var,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X )
= X ) ) ).
% eval_var
thf(fact_21_eval__poly__of__const,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_of_const_a_b @ r @ X ) @ Y )
= X ) ) ).
% eval_poly_of_const
thf(fact_22_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_23_finprod__one,axiom,
! [A: set_a] :
( ( finpro205304725090349623_a_b_a @ r
@ ^ [I: a] : ( one_a_ring_ext_a_b @ r )
@ A )
= ( one_a_ring_ext_a_b @ r ) ) ).
% finprod_one
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_nat,F: nat > a] :
( ~ ( finite_finite_nat @ A )
=> ( ( finpro1280035270526425175_b_nat @ r @ F @ A )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_26_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_27_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
lagran9092808442999052491ux_a_b = lagran9092808442999052491ux_a_b ).
% ring.lagrange_basis_polynomial_aux.cong
thf(fact_28_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
lagran3534788790333317459t_unit = lagran3534788790333317459t_unit ).
% ring.lagrange_basis_polynomial_aux.cong
thf(fact_29_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B: set_nat] : ( ord_less_eq_set_nat @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_30_finite__Collect__subsets,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B: set_a] : ( ord_less_eq_set_a @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_31_finite__Collect__subsets,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( finite5282473924520328461list_a
@ ( collect_set_list_a
@ ^ [B: set_list_a] : ( ord_le8861187494160871172list_a @ B @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_32_finite__Diff,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_33_finite__Diff,axiom,
! [A: set_a,B2: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( minus_minus_set_a @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_34_finite__Diff,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_35_finite__Diff2,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_36_finite__Diff2,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A @ B2 ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_Diff2
thf(fact_37_finite__Diff2,axiom,
! [B2: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A @ B2 ) )
= ( finite_finite_list_a @ A ) ) ) ).
% finite_Diff2
thf(fact_38_finprod__mono__neutral__cong__right,axiom,
! [B2: set_set_list_a,A: set_set_list_a,G: set_list_a > a,H: set_list_a > a] :
( ( finite5282473924520328461list_a @ B2 )
=> ( ( ord_le8877086941679407844list_a @ A @ B2 )
=> ( ! [I2: set_list_a] :
( ( member_set_list_a @ I2 @ ( minus_4782336368215558443list_a @ B2 @ A ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ B2
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3826550488720007709list_a @ r @ G @ B2 )
= ( finpro3826550488720007709list_a @ r @ H @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_39_finprod__mono__neutral__cong__right,axiom,
! [B2: set_list_a_a,A: set_list_a_a,G: ( list_a > a ) > a,H: ( list_a > a ) > a] :
( ( finite2458174228029419510st_a_a @ B2 )
=> ( ( ord_le6942402695062981877st_a_a @ A @ B2 )
=> ( ! [I2: list_a > a] :
( ( member_list_a_a @ I2 @ ( minus_921748639838131438st_a_a @ B2 @ A ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: list_a > a] :
( ( member_list_a_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_a_a_a @ G
@ ( pi_list_a_a_a @ B2
@ ^ [Uu: list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5422611492341532390st_a_a @ r @ G @ B2 )
= ( finpro5422611492341532390st_a_a @ r @ H @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_40_finprod__mono__neutral__cong__right,axiom,
! [B2: set_set_list_a_a,A: set_set_list_a_a,G: ( set_list_a > a ) > a,H: ( set_list_a > a ) > a] :
( ( finite6385009043124570134st_a_a @ B2 )
=> ( ( ord_le4799719167512954133st_a_a @ A @ B2 )
=> ( ! [I2: set_list_a > a] :
( ( member_set_list_a_a @ I2 @ ( minus_5613498140476352782st_a_a @ B2 @ A ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member969817812316227871_a_a_a @ G
@ ( pi_set_list_a_a_a @ B2
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4938371440467910406st_a_a @ r @ G @ B2 )
= ( finpro4938371440467910406st_a_a @ r @ H @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_41_finprod__mono__neutral__cong__right,axiom,
! [B2: set_nat_list_a,A: set_nat_list_a,G: ( nat > list_a ) > a,H: ( nat > list_a ) > a] :
( ( finite7630042315537210004list_a @ B2 )
=> ( ( ord_le2145805922479659755list_a @ A @ B2 )
=> ( ! [I2: nat > list_a] :
( ( member_nat_list_a @ I2 @ ( minus_4169782841487898290list_a @ B2 @ A ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_list_a_a @ G
@ ( pi_nat_list_a_a @ B2
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4838020199848830884list_a @ r @ G @ B2 )
= ( finpro4838020199848830884list_a @ r @ H @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_42_finprod__mono__neutral__cong__right,axiom,
! [B2: set_nat_a,A: set_nat_a,G: ( nat > a ) > a,H: ( nat > a ) > a] :
( ( finite_finite_nat_a @ B2 )
=> ( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( ! [I2: nat > a] :
( ( member_nat_a @ I2 @ ( minus_490503922182417452_nat_a @ B2 @ A ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5839458686994656414_nat_a @ r @ G @ B2 )
= ( finpro5839458686994656414_nat_a @ r @ H @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_43_finprod__mono__neutral__cong__right,axiom,
! [B2: set_nat,A: set_nat,G: nat > a,H: nat > a] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ B2 @ A ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r @ G @ B2 )
= ( finpro1280035270526425175_b_nat @ r @ H @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_44_finprod__mono__neutral__cong__right,axiom,
! [B2: set_a,A: set_a,G: a > a,H: a > a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ B2 @ A ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ G @ B2 )
= ( finpro205304725090349623_a_b_a @ r @ H @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_45_finprod__mono__neutral__cong__right,axiom,
! [B2: set_list_a,A: set_list_a,G: list_a > a,H: list_a > a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ ( minus_646659088055828811list_a @ B2 @ A ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r @ G @ B2 )
= ( finpro6052973074229812797list_a @ r @ H @ A ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_right
thf(fact_46_finprod__mono__neutral__cong__left,axiom,
! [B2: set_set_list_a,A: set_set_list_a,H: set_list_a > a,G: set_list_a > a] :
( ( finite5282473924520328461list_a @ B2 )
=> ( ( ord_le8877086941679407844list_a @ A @ B2 )
=> ( ! [I2: set_list_a] :
( ( member_set_list_a @ I2 @ ( minus_4782336368215558443list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_set_list_a_a @ H
@ ( pi_set_list_a_a @ B2
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3826550488720007709list_a @ r @ G @ A )
= ( finpro3826550488720007709list_a @ r @ H @ B2 ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_47_finprod__mono__neutral__cong__left,axiom,
! [B2: set_list_a_a,A: set_list_a_a,H: ( list_a > a ) > a,G: ( list_a > a ) > a] :
( ( finite2458174228029419510st_a_a @ B2 )
=> ( ( ord_le6942402695062981877st_a_a @ A @ B2 )
=> ( ! [I2: list_a > a] :
( ( member_list_a_a @ I2 @ ( minus_921748639838131438st_a_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: list_a > a] :
( ( member_list_a_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_a_a_a @ H
@ ( pi_list_a_a_a @ B2
@ ^ [Uu: list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5422611492341532390st_a_a @ r @ G @ A )
= ( finpro5422611492341532390st_a_a @ r @ H @ B2 ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_48_finprod__mono__neutral__cong__left,axiom,
! [B2: set_set_list_a_a,A: set_set_list_a_a,H: ( set_list_a > a ) > a,G: ( set_list_a > a ) > a] :
( ( finite6385009043124570134st_a_a @ B2 )
=> ( ( ord_le4799719167512954133st_a_a @ A @ B2 )
=> ( ! [I2: set_list_a > a] :
( ( member_set_list_a_a @ I2 @ ( minus_5613498140476352782st_a_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member969817812316227871_a_a_a @ H
@ ( pi_set_list_a_a_a @ B2
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4938371440467910406st_a_a @ r @ G @ A )
= ( finpro4938371440467910406st_a_a @ r @ H @ B2 ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_49_finprod__mono__neutral__cong__left,axiom,
! [B2: set_nat_list_a,A: set_nat_list_a,H: ( nat > list_a ) > a,G: ( nat > list_a ) > a] :
( ( finite7630042315537210004list_a @ B2 )
=> ( ( ord_le2145805922479659755list_a @ A @ B2 )
=> ( ! [I2: nat > list_a] :
( ( member_nat_list_a @ I2 @ ( minus_4169782841487898290list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_list_a_a @ H
@ ( pi_nat_list_a_a @ B2
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4838020199848830884list_a @ r @ G @ A )
= ( finpro4838020199848830884list_a @ r @ H @ B2 ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_50_finprod__mono__neutral__cong__left,axiom,
! [B2: set_nat_a,A: set_nat_a,H: ( nat > a ) > a,G: ( nat > a ) > a] :
( ( finite_finite_nat_a @ B2 )
=> ( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( ! [I2: nat > a] :
( ( member_nat_a @ I2 @ ( minus_490503922182417452_nat_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a_a @ H
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5839458686994656414_nat_a @ r @ G @ A )
= ( finpro5839458686994656414_nat_a @ r @ H @ B2 ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_51_finprod__mono__neutral__cong__left,axiom,
! [B2: set_nat,A: set_nat,H: nat > a,G: nat > a] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a @ H
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r @ G @ A )
= ( finpro1280035270526425175_b_nat @ r @ H @ B2 ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_52_finprod__mono__neutral__cong__left,axiom,
! [B2: set_a,A: set_a,H: a > a,G: a > a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_a @ H
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ G @ A )
= ( finpro205304725090349623_a_b_a @ r @ H @ B2 ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_53_finprod__mono__neutral__cong__left,axiom,
! [B2: set_list_a,A: set_list_a,H: list_a > a,G: list_a > a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ ( minus_646659088055828811list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_a_a @ H
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r @ G @ A )
= ( finpro6052973074229812797list_a @ r @ H @ B2 ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong_left
thf(fact_54_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_55_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_56_a__l__coset__subset__G,axiom,
! [H2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_57_subset__Idl__subset,axiom,
! [I3: set_a,H2: set_a] :
( ( ord_less_eq_set_a @ I3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H2 @ I3 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H2 ) @ ( genideal_a_b @ r @ I3 ) ) ) ) ).
% subset_Idl_subset
thf(fact_58_genideal__self,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ r @ S ) ) ) ).
% genideal_self
thf(fact_59_cgenideal__is__principalideal,axiom,
! [I4: a] :
( ( member_a @ I4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I4 ) @ r ) ) ).
% cgenideal_is_principalideal
thf(fact_60_cgenideal__self,axiom,
! [I4: a] :
( ( member_a @ I4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I4 @ ( cgenid547466209912283029xt_a_b @ r @ I4 ) ) ) ).
% cgenideal_self
thf(fact_61_finprod__closed,axiom,
! [F: list_a > a,A: set_list_a] :
( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_a @ ( finpro6052973074229812797list_a @ r @ F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% finprod_closed
thf(fact_62_finprod__closed,axiom,
! [F: set_list_a > a,A: set_set_list_a] :
( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_a @ ( finpro3826550488720007709list_a @ r @ F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% finprod_closed
thf(fact_63_finprod__closed,axiom,
! [F: nat > a,A: set_nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_a @ ( finpro1280035270526425175_b_nat @ r @ F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% finprod_closed
thf(fact_64_finprod__closed,axiom,
! [F: a > a,A: set_a] :
( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_a @ ( finpro205304725090349623_a_b_a @ r @ F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% finprod_closed
thf(fact_65_finprod__cong_H,axiom,
! [A: set_list_a_a,B2: set_list_a_a,G: ( list_a > a ) > a,F: ( list_a > a ) > a] :
( ( A = B2 )
=> ( ( member_list_a_a_a @ G
@ ( pi_list_a_a_a @ B2
@ ^ [Uu: list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: list_a > a] :
( ( member_list_a_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finpro5422611492341532390st_a_a @ r @ F @ A )
= ( finpro5422611492341532390st_a_a @ r @ G @ B2 ) ) ) ) ) ).
% finprod_cong'
thf(fact_66_finprod__cong_H,axiom,
! [A: set_set_list_a_a,B2: set_set_list_a_a,G: ( set_list_a > a ) > a,F: ( set_list_a > a ) > a] :
( ( A = B2 )
=> ( ( member969817812316227871_a_a_a @ G
@ ( pi_set_list_a_a_a @ B2
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: set_list_a > a] :
( ( member_set_list_a_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finpro4938371440467910406st_a_a @ r @ F @ A )
= ( finpro4938371440467910406st_a_a @ r @ G @ B2 ) ) ) ) ) ).
% finprod_cong'
thf(fact_67_finprod__cong_H,axiom,
! [A: set_nat_list_a,B2: set_nat_list_a,G: ( nat > list_a ) > a,F: ( nat > list_a ) > a] :
( ( A = B2 )
=> ( ( member_nat_list_a_a @ G
@ ( pi_nat_list_a_a @ B2
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: nat > list_a] :
( ( member_nat_list_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finpro4838020199848830884list_a @ r @ F @ A )
= ( finpro4838020199848830884list_a @ r @ G @ B2 ) ) ) ) ) ).
% finprod_cong'
thf(fact_68_finprod__cong_H,axiom,
! [A: set_nat_a,B2: set_nat_a,G: ( nat > a ) > a,F: ( nat > a ) > a] :
( ( A = B2 )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: nat > a] :
( ( member_nat_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finpro5839458686994656414_nat_a @ r @ F @ A )
= ( finpro5839458686994656414_nat_a @ r @ G @ B2 ) ) ) ) ) ).
% finprod_cong'
thf(fact_69_finprod__cong_H,axiom,
! [A: set_list_a,B2: set_list_a,G: list_a > a,F: list_a > a] :
( ( A = B2 )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finpro6052973074229812797list_a @ r @ F @ A )
= ( finpro6052973074229812797list_a @ r @ G @ B2 ) ) ) ) ) ).
% finprod_cong'
thf(fact_70_finprod__cong_H,axiom,
! [A: set_set_list_a,B2: set_set_list_a,G: set_list_a > a,F: set_list_a > a] :
( ( A = B2 )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ B2
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: set_list_a] :
( ( member_set_list_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finpro3826550488720007709list_a @ r @ F @ A )
= ( finpro3826550488720007709list_a @ r @ G @ B2 ) ) ) ) ) ).
% finprod_cong'
thf(fact_71_finprod__cong_H,axiom,
! [A: set_nat,B2: set_nat,G: nat > a,F: nat > a] :
( ( A = B2 )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r @ F @ A )
= ( finpro1280035270526425175_b_nat @ r @ G @ B2 ) ) ) ) ) ).
% finprod_cong'
thf(fact_72_finprod__cong_H,axiom,
! [A: set_a,B2: set_a,G: a > a,F: a > a] :
( ( A = B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [I2: a] :
( ( member_a @ I2 @ B2 )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( finpro205304725090349623_a_b_a @ r @ G @ B2 ) ) ) ) ) ).
% finprod_cong'
thf(fact_73_finite__Collect__conjI,axiom,
! [P2: set_list_a > $o,Q2: set_list_a > $o] :
( ( ( finite5282473924520328461list_a @ ( collect_set_list_a @ P2 ) )
| ( finite5282473924520328461list_a @ ( collect_set_list_a @ Q2 ) ) )
=> ( finite5282473924520328461list_a
@ ( collect_set_list_a
@ ^ [X3: set_list_a] :
( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_74_finite__Collect__conjI,axiom,
! [P2: set_a > $o,Q2: set_a > $o] :
( ( ( finite_finite_set_a @ ( collect_set_a @ P2 ) )
| ( finite_finite_set_a @ ( collect_set_a @ Q2 ) ) )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_75_finite__Collect__conjI,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P2 ) )
| ( finite_finite_a @ ( collect_a @ Q2 ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_76_finite__Collect__conjI,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
| ( finite_finite_list_a @ ( collect_list_a @ Q2 ) ) )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X3: list_a] :
( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_77_finite__Collect__conjI,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
| ( finite_finite_nat @ ( collect_nat @ Q2 ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_78_finite__Collect__disjI,axiom,
! [P2: set_list_a > $o,Q2: set_list_a > $o] :
( ( finite5282473924520328461list_a
@ ( collect_set_list_a
@ ^ [X3: set_list_a] :
( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) )
= ( ( finite5282473924520328461list_a @ ( collect_set_list_a @ P2 ) )
& ( finite5282473924520328461list_a @ ( collect_set_list_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_79_finite__Collect__disjI,axiom,
! [P2: set_a > $o,Q2: set_a > $o] :
( ( finite_finite_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) )
= ( ( finite_finite_set_a @ ( collect_set_a @ P2 ) )
& ( finite_finite_set_a @ ( collect_set_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_80_finite__Collect__disjI,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P2 ) )
& ( finite_finite_a @ ( collect_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_81_finite__Collect__disjI,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X3: list_a] :
( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) )
= ( ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
& ( finite_finite_list_a @ ( collect_list_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_82_finite__Collect__disjI,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
& ( finite_finite_nat @ ( collect_nat @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_83_finprod__singleton,axiom,
! [I4: set_list_a,A: set_set_list_a,F: set_list_a > a] :
( ( member_set_list_a @ I4 @ A )
=> ( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3826550488720007709list_a @ r
@ ^ [J: set_list_a] : ( if_a @ ( I4 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton
thf(fact_84_finprod__singleton,axiom,
! [I4: list_a > a,A: set_list_a_a,F: ( list_a > a ) > a] :
( ( member_list_a_a @ I4 @ A )
=> ( ( finite2458174228029419510st_a_a @ A )
=> ( ( member_list_a_a_a @ F
@ ( pi_list_a_a_a @ A
@ ^ [Uu: list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5422611492341532390st_a_a @ r
@ ^ [J: list_a > a] : ( if_a @ ( I4 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton
thf(fact_85_finprod__singleton,axiom,
! [I4: set_list_a > a,A: set_set_list_a_a,F: ( set_list_a > a ) > a] :
( ( member_set_list_a_a @ I4 @ A )
=> ( ( finite6385009043124570134st_a_a @ A )
=> ( ( member969817812316227871_a_a_a @ F
@ ( pi_set_list_a_a_a @ A
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4938371440467910406st_a_a @ r
@ ^ [J: set_list_a > a] : ( if_a @ ( I4 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton
thf(fact_86_finprod__singleton,axiom,
! [I4: nat > list_a,A: set_nat_list_a,F: ( nat > list_a ) > a] :
( ( member_nat_list_a @ I4 @ A )
=> ( ( finite7630042315537210004list_a @ A )
=> ( ( member_nat_list_a_a @ F
@ ( pi_nat_list_a_a @ A
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4838020199848830884list_a @ r
@ ^ [J: nat > list_a] : ( if_a @ ( I4 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton
thf(fact_87_finprod__singleton,axiom,
! [I4: nat > a,A: set_nat_a,F: ( nat > a ) > a] :
( ( member_nat_a @ I4 @ A )
=> ( ( finite_finite_nat_a @ A )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5839458686994656414_nat_a @ r
@ ^ [J: nat > a] : ( if_a @ ( I4 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton
thf(fact_88_finprod__singleton,axiom,
! [I4: list_a,A: set_list_a,F: list_a > a] :
( ( member_list_a @ I4 @ A )
=> ( ( finite_finite_list_a @ A )
=> ( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r
@ ^ [J: list_a] : ( if_a @ ( I4 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton
thf(fact_89_finprod__singleton,axiom,
! [I4: nat,A: set_nat,F: nat > a] :
( ( member_nat @ I4 @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r
@ ^ [J: nat] : ( if_a @ ( I4 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton
thf(fact_90_finprod__singleton,axiom,
! [I4: a,A: set_a,F: a > a] :
( ( member_a @ I4 @ A )
=> ( ( finite_finite_a @ A )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r
@ ^ [J: a] : ( if_a @ ( I4 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton
thf(fact_91_finprod__singleton__swap,axiom,
! [I4: set_list_a,A: set_set_list_a,F: set_list_a > a] :
( ( member_set_list_a @ I4 @ A )
=> ( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3826550488720007709list_a @ r
@ ^ [J: set_list_a] : ( if_a @ ( J = I4 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_92_finprod__singleton__swap,axiom,
! [I4: list_a > a,A: set_list_a_a,F: ( list_a > a ) > a] :
( ( member_list_a_a @ I4 @ A )
=> ( ( finite2458174228029419510st_a_a @ A )
=> ( ( member_list_a_a_a @ F
@ ( pi_list_a_a_a @ A
@ ^ [Uu: list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5422611492341532390st_a_a @ r
@ ^ [J: list_a > a] : ( if_a @ ( J = I4 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_93_finprod__singleton__swap,axiom,
! [I4: set_list_a > a,A: set_set_list_a_a,F: ( set_list_a > a ) > a] :
( ( member_set_list_a_a @ I4 @ A )
=> ( ( finite6385009043124570134st_a_a @ A )
=> ( ( member969817812316227871_a_a_a @ F
@ ( pi_set_list_a_a_a @ A
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4938371440467910406st_a_a @ r
@ ^ [J: set_list_a > a] : ( if_a @ ( J = I4 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_94_finprod__singleton__swap,axiom,
! [I4: nat > list_a,A: set_nat_list_a,F: ( nat > list_a ) > a] :
( ( member_nat_list_a @ I4 @ A )
=> ( ( finite7630042315537210004list_a @ A )
=> ( ( member_nat_list_a_a @ F
@ ( pi_nat_list_a_a @ A
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4838020199848830884list_a @ r
@ ^ [J: nat > list_a] : ( if_a @ ( J = I4 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_95_finprod__singleton__swap,axiom,
! [I4: nat > a,A: set_nat_a,F: ( nat > a ) > a] :
( ( member_nat_a @ I4 @ A )
=> ( ( finite_finite_nat_a @ A )
=> ( ( member_nat_a_a @ F
@ ( pi_nat_a_a @ A
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5839458686994656414_nat_a @ r
@ ^ [J: nat > a] : ( if_a @ ( J = I4 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_96_finprod__singleton__swap,axiom,
! [I4: list_a,A: set_list_a,F: list_a > a] :
( ( member_list_a @ I4 @ A )
=> ( ( finite_finite_list_a @ A )
=> ( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r
@ ^ [J: list_a] : ( if_a @ ( J = I4 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_97_finprod__singleton__swap,axiom,
! [I4: nat,A: set_nat,F: nat > a] :
( ( member_nat @ I4 @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r
@ ^ [J: nat] : ( if_a @ ( J = I4 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_98_finprod__singleton__swap,axiom,
! [I4: a,A: set_a,F: a > a] :
( ( member_a @ I4 @ A )
=> ( ( finite_finite_a @ A )
=> ( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r
@ ^ [J: a] : ( if_a @ ( J = I4 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
@ A )
= ( F @ I4 ) ) ) ) ) ).
% finprod_singleton_swap
thf(fact_99_not__finite__existsD,axiom,
! [P2: set_list_a > $o] :
( ~ ( finite5282473924520328461list_a @ ( collect_set_list_a @ P2 ) )
=> ? [X_1: set_list_a] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_100_not__finite__existsD,axiom,
! [P2: set_a > $o] :
( ~ ( finite_finite_set_a @ ( collect_set_a @ P2 ) )
=> ? [X_1: set_a] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_101_not__finite__existsD,axiom,
! [P2: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P2 ) )
=> ? [X_1: a] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_102_not__finite__existsD,axiom,
! [P2: list_a > $o] :
( ~ ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
=> ? [X_1: list_a] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_103_not__finite__existsD,axiom,
! [P2: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ? [X_1: nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_104_mem__Collect__eq,axiom,
! [A2: list_a > a,P2: ( list_a > a ) > $o] :
( ( member_list_a_a @ A2 @ ( collect_list_a_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_105_mem__Collect__eq,axiom,
! [A2: set_list_a > a,P2: ( set_list_a > a ) > $o] :
( ( member_set_list_a_a @ A2 @ ( collect_set_list_a_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_106_mem__Collect__eq,axiom,
! [A2: nat > list_a,P2: ( nat > list_a ) > $o] :
( ( member_nat_list_a @ A2 @ ( collect_nat_list_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_107_mem__Collect__eq,axiom,
! [A2: nat > a,P2: ( nat > a ) > $o] :
( ( member_nat_a @ A2 @ ( collect_nat_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_108_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_109_mem__Collect__eq,axiom,
! [A2: a,P2: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_110_mem__Collect__eq,axiom,
! [A2: set_list_a,P2: set_list_a > $o] :
( ( member_set_list_a @ A2 @ ( collect_set_list_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_111_mem__Collect__eq,axiom,
! [A2: set_a,P2: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_112_mem__Collect__eq,axiom,
! [A2: nat,P2: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_113_Collect__mem__eq,axiom,
! [A: set_list_a_a] :
( ( collect_list_a_a
@ ^ [X3: list_a > a] : ( member_list_a_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
! [A: set_set_list_a_a] :
( ( collect_set_list_a_a
@ ^ [X3: set_list_a > a] : ( member_set_list_a_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_115_Collect__mem__eq,axiom,
! [A: set_nat_list_a] :
( ( collect_nat_list_a
@ ^ [X3: nat > list_a] : ( member_nat_list_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_116_Collect__mem__eq,axiom,
! [A: set_nat_a] :
( ( collect_nat_a
@ ^ [X3: nat > a] : ( member_nat_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_117_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_118_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_119_Collect__mem__eq,axiom,
! [A: set_set_list_a] :
( ( collect_set_list_a
@ ^ [X3: set_list_a] : ( member_set_list_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_120_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_121_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_122_Collect__cong,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ! [X2: list_a] :
( ( P2 @ X2 )
= ( Q2 @ X2 ) )
=> ( ( collect_list_a @ P2 )
= ( collect_list_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_123_Collect__cong,axiom,
! [P2: a > $o,Q2: a > $o] :
( ! [X2: a] :
( ( P2 @ X2 )
= ( Q2 @ X2 ) )
=> ( ( collect_a @ P2 )
= ( collect_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_124_Collect__cong,axiom,
! [P2: set_list_a > $o,Q2: set_list_a > $o] :
( ! [X2: set_list_a] :
( ( P2 @ X2 )
= ( Q2 @ X2 ) )
=> ( ( collect_set_list_a @ P2 )
= ( collect_set_list_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_125_Collect__cong,axiom,
! [P2: set_a > $o,Q2: set_a > $o] :
( ! [X2: set_a] :
( ( P2 @ X2 )
= ( Q2 @ X2 ) )
=> ( ( collect_set_a @ P2 )
= ( collect_set_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_126_Collect__cong,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ! [X2: nat] :
( ( P2 @ X2 )
= ( Q2 @ X2 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_127_pigeonhole__infinite__rel,axiom,
! [A: set_a,B2: set_a,R: a > a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_128_pigeonhole__infinite__rel,axiom,
! [A: set_a,B2: set_nat,R: a > nat > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_129_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B2: set_a,R: nat > a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_130_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B2: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_131_pigeonhole__infinite__rel,axiom,
! [A: set_set_a,B2: set_a,R: set_a > a > $o] :
( ~ ( finite_finite_set_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B2 )
& ~ ( finite_finite_set_a
@ ( collect_set_a
@ ^ [A3: set_a] :
( ( member_set_a @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_132_pigeonhole__infinite__rel,axiom,
! [A: set_set_a,B2: set_nat,R: set_a > nat > $o] :
( ~ ( finite_finite_set_a @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_set_a
@ ( collect_set_a
@ ^ [A3: set_a] :
( ( member_set_a @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_133_pigeonhole__infinite__rel,axiom,
! [A: set_a,B2: set_list_a,R: a > list_a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A3: a] :
( ( member_a @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_134_pigeonhole__infinite__rel,axiom,
! [A: set_list_a,B2: set_a,R: list_a > a > $o] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ B2 )
& ~ ( finite_finite_list_a
@ ( collect_list_a
@ ^ [A3: list_a] :
( ( member_list_a @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_135_pigeonhole__infinite__rel,axiom,
! [A: set_list_a,B2: set_nat,R: list_a > nat > $o] :
( ~ ( finite_finite_list_a @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ~ ( finite_finite_list_a
@ ( collect_list_a
@ ^ [A3: list_a] :
( ( member_list_a @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_136_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B2: set_list_a,R: nat > list_a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ B2 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R @ A3 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_137_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_138_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_139_finite__has__minimal2,axiom,
! [A: set_set_list_a,A2: set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a @ A2 @ A )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
& ( ord_le8861187494160871172list_a @ X2 @ A2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_140_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A2 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_141_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ A2 @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_142_finite__has__maximal2,axiom,
! [A: set_set_list_a,A2: set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a @ A2 @ A )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
& ( ord_le8861187494160871172list_a @ A2 @ X2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_143_rev__finite__subset,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_144_rev__finite__subset,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_145_rev__finite__subset,axiom,
! [B2: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( finite_finite_list_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_146_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_147_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_148_infinite__super,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_super
thf(fact_149_finite__subset,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_150_finite__subset,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_151_finite__subset,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( finite_finite_list_a @ B2 )
=> ( finite_finite_list_a @ A ) ) ) ).
% finite_subset
thf(fact_152_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_153_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_154_Diff__infinite__finite,axiom,
! [T: set_list_a,S: set_list_a] :
( ( finite_finite_list_a @ T )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_155_finprod__mono__neutral__cong,axiom,
! [B2: set_set_list_a,A: set_set_list_a,H: set_list_a > a,G: set_list_a > a] :
( ( finite5282473924520328461list_a @ B2 )
=> ( ( finite5282473924520328461list_a @ A )
=> ( ! [I2: set_list_a] :
( ( member_set_list_a @ I2 @ ( minus_4782336368215558443list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I2: set_list_a] :
( ( member_set_list_a @ I2 @ ( minus_4782336368215558443list_a @ A @ B2 ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ ( inf_in4657809108759609906list_a @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_set_list_a_a @ H
@ ( pi_set_list_a_a @ B2
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3826550488720007709list_a @ r @ G @ A )
= ( finpro3826550488720007709list_a @ r @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_156_finprod__mono__neutral__cong,axiom,
! [B2: set_list_a_a,A: set_list_a_a,H: ( list_a > a ) > a,G: ( list_a > a ) > a] :
( ( finite2458174228029419510st_a_a @ B2 )
=> ( ( finite2458174228029419510st_a_a @ A )
=> ( ! [I2: list_a > a] :
( ( member_list_a_a @ I2 @ ( minus_921748639838131438st_a_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I2: list_a > a] :
( ( member_list_a_a @ I2 @ ( minus_921748639838131438st_a_a @ A @ B2 ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: list_a > a] :
( ( member_list_a_a @ X2 @ ( inf_inf_set_list_a_a @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_a_a_a @ G
@ ( pi_list_a_a_a @ A
@ ^ [Uu: list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a_a_a @ H
@ ( pi_list_a_a_a @ B2
@ ^ [Uu: list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5422611492341532390st_a_a @ r @ G @ A )
= ( finpro5422611492341532390st_a_a @ r @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_157_finprod__mono__neutral__cong,axiom,
! [B2: set_set_list_a_a,A: set_set_list_a_a,H: ( set_list_a > a ) > a,G: ( set_list_a > a ) > a] :
( ( finite6385009043124570134st_a_a @ B2 )
=> ( ( finite6385009043124570134st_a_a @ A )
=> ( ! [I2: set_list_a > a] :
( ( member_set_list_a_a @ I2 @ ( minus_5613498140476352782st_a_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I2: set_list_a > a] :
( ( member_set_list_a_a @ I2 @ ( minus_5613498140476352782st_a_a @ A @ B2 ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ ( inf_in6568206481208318535st_a_a @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member969817812316227871_a_a_a @ G
@ ( pi_set_list_a_a_a @ A
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member969817812316227871_a_a_a @ H
@ ( pi_set_list_a_a_a @ B2
@ ^ [Uu: set_list_a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4938371440467910406st_a_a @ r @ G @ A )
= ( finpro4938371440467910406st_a_a @ r @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_158_finprod__mono__neutral__cong,axiom,
! [B2: set_nat_list_a,A: set_nat_list_a,H: ( nat > list_a ) > a,G: ( nat > list_a ) > a] :
( ( finite7630042315537210004list_a @ B2 )
=> ( ( finite7630042315537210004list_a @ A )
=> ( ! [I2: nat > list_a] :
( ( member_nat_list_a @ I2 @ ( minus_4169782841487898290list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I2: nat > list_a] :
( ( member_nat_list_a @ I2 @ ( minus_4169782841487898290list_a @ A @ B2 ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ ( inf_in6652419485960844601list_a @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_list_a_a @ G
@ ( pi_nat_list_a_a @ A
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_list_a_a @ H
@ ( pi_nat_list_a_a @ B2
@ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro4838020199848830884list_a @ r @ G @ A )
= ( finpro4838020199848830884list_a @ r @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_159_finprod__mono__neutral__cong,axiom,
! [B2: set_nat_a,A: set_nat_a,H: ( nat > a ) > a,G: ( nat > a ) > a] :
( ( finite_finite_nat_a @ B2 )
=> ( ( finite_finite_nat_a @ A )
=> ( ! [I2: nat > a] :
( ( member_nat_a @ I2 @ ( minus_490503922182417452_nat_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I2: nat > a] :
( ( member_nat_a @ I2 @ ( minus_490503922182417452_nat_a @ A @ B2 ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ ( inf_inf_set_nat_a @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a_a @ G
@ ( pi_nat_a_a @ A
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_a_a @ H
@ ( pi_nat_a_a @ B2
@ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro5839458686994656414_nat_a @ r @ G @ A )
= ( finpro5839458686994656414_nat_a @ r @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_160_finprod__mono__neutral__cong,axiom,
! [B2: set_nat,A: set_nat,H: nat > a,G: nat > a] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ A )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ ( minus_minus_set_nat @ A @ B2 ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_a @ H
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r @ G @ A )
= ( finpro1280035270526425175_b_nat @ r @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_161_finprod__mono__neutral__cong,axiom,
! [B2: set_a,A: set_a,H: a > a,G: a > a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ A )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I2: a] :
( ( member_a @ I2 @ ( minus_minus_set_a @ A @ B2 ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( inf_inf_set_a @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a @ H
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ G @ A )
= ( finpro205304725090349623_a_b_a @ r @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_162_finprod__mono__neutral__cong,axiom,
! [B2: set_list_a,A: set_list_a,H: list_a > a,G: list_a > a] :
( ( finite_finite_list_a @ B2 )
=> ( ( finite_finite_list_a @ A )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ ( minus_646659088055828811list_a @ B2 @ A ) )
=> ( ( H @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [I2: list_a] :
( ( member_list_a @ I2 @ ( minus_646659088055828811list_a @ A @ B2 ) )
=> ( ( G @ I2 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( inf_inf_set_list_a @ A @ B2 ) )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a_a @ H
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r @ G @ A )
= ( finpro6052973074229812797list_a @ r @ H @ B2 ) ) ) ) ) ) ) ) ) ).
% finprod_mono_neutral_cong
thf(fact_163_noetherian__ringI,axiom,
( ! [I5: set_a] :
( ( ideal_a_b @ I5 @ r )
=> ? [A4: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( finite_finite_a @ A4 )
& ( I5
= ( genideal_a_b @ r @ A4 ) ) ) )
=> ( ring_n3639167112692572309ng_a_b @ r ) ) ).
% noetherian_ringI
thf(fact_164_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_165_a__lcos__m__assoc,axiom,
! [M: set_a,G: a,H: a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H @ M ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H ) @ M ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_166_finetely__gen,axiom,
! [I3: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ? [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( finite_finite_a @ A5 )
& ( I3
= ( genideal_a_b @ r @ A5 ) ) ) ) ).
% finetely_gen
thf(fact_167_finprod__zero__iff,axiom,
! [A: set_list_a_a,F: ( list_a > a ) > a] :
( ( finite2458174228029419510st_a_a @ A )
=> ( ! [A6: list_a > a] :
( ( member_list_a_a @ A6 @ A )
=> ( member_a @ ( F @ A6 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro5422611492341532390st_a_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X3: list_a > a] :
( ( member_list_a_a @ X3 @ A )
& ( ( F @ X3 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_168_finprod__zero__iff,axiom,
! [A: set_set_list_a_a,F: ( set_list_a > a ) > a] :
( ( finite6385009043124570134st_a_a @ A )
=> ( ! [A6: set_list_a > a] :
( ( member_set_list_a_a @ A6 @ A )
=> ( member_a @ ( F @ A6 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro4938371440467910406st_a_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X3: set_list_a > a] :
( ( member_set_list_a_a @ X3 @ A )
& ( ( F @ X3 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_169_finprod__zero__iff,axiom,
! [A: set_nat_list_a,F: ( nat > list_a ) > a] :
( ( finite7630042315537210004list_a @ A )
=> ( ! [A6: nat > list_a] :
( ( member_nat_list_a @ A6 @ A )
=> ( member_a @ ( F @ A6 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro4838020199848830884list_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X3: nat > list_a] :
( ( member_nat_list_a @ X3 @ A )
& ( ( F @ X3 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_170_finprod__zero__iff,axiom,
! [A: set_nat_a,F: ( nat > a ) > a] :
( ( finite_finite_nat_a @ A )
=> ( ! [A6: nat > a] :
( ( member_nat_a @ A6 @ A )
=> ( member_a @ ( F @ A6 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro5839458686994656414_nat_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X3: nat > a] :
( ( member_nat_a @ X3 @ A )
& ( ( F @ X3 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_171_finprod__zero__iff,axiom,
! [A: set_list_a,F: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ! [A6: list_a] :
( ( member_list_a @ A6 @ A )
=> ( member_a @ ( F @ A6 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro6052973074229812797list_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ A )
& ( ( F @ X3 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_172_finprod__zero__iff,axiom,
! [A: set_nat,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ( member_a @ ( F @ A6 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro1280035270526425175_b_nat @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( F @ X3 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_173_finprod__zero__iff,axiom,
! [A: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ! [A6: a] :
( ( member_a @ A6 @ A )
=> ( member_a @ ( F @ A6 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro205304725090349623_a_b_a @ r @ F @ A )
= ( zero_a_b @ r ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A )
& ( ( F @ X3 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_174_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_175_eval__in__carrier,axiom,
! [P: list_a,X: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier
thf(fact_176_up__smult__closed,axiom,
! [A2: a,P: nat > a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I: nat] : ( mult_a_ring_ext_a_b @ r @ A2 @ ( P @ I ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_smult_closed
thf(fact_177_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_178_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_179_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_180_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_181_to__contain__is__to__divide,axiom,
! [A2: a,B3: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( cgenid547466209912283029xt_a_b @ r @ B3 ) @ ( cgenid547466209912283029xt_a_b @ r @ A2 ) )
= ( factor8216151070175719842xt_a_b @ r @ A2 @ B3 ) ) ) ) ).
% to_contain_is_to_divide
thf(fact_182_i__intersect,axiom,
! [I3: set_a,J2: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ( ( ideal_a_b @ J2 @ r )
=> ( ideal_a_b @ ( inf_inf_set_a @ I3 @ J2 ) @ r ) ) ) ).
% i_intersect
thf(fact_183_subalgebra__inter,axiom,
! [K: set_a,V: set_a,V2: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V2 @ r )
=> ( embedd9027525575939734154ra_a_b @ K @ ( inf_inf_set_a @ V @ V2 ) @ r ) ) ) ).
% subalgebra_inter
thf(fact_184_a__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_185_a__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) ) ) ) ).
% a_comm
thf(fact_186_a__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
= ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_187_add_Or__cancel,axiom,
! [A2: a,C: a,B3: a] :
( ( ( add_a_b @ r @ A2 @ C )
= ( add_a_b @ r @ B3 @ C ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B3 ) ) ) ) ) ).
% add.r_cancel
thf(fact_188_add_Ol__cancel,axiom,
! [C: a,A2: a,B3: a] :
( ( ( add_a_b @ r @ C @ A2 )
= ( add_a_b @ r @ C @ B3 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B3 ) ) ) ) ) ).
% add.l_cancel
thf(fact_189_m__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_190_m__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% m_comm
thf(fact_191_m__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_192_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_193_divides__trans,axiom,
! [A2: a,B3: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A2 @ B3 )
=> ( ( factor8216151070175719842xt_a_b @ r @ B3 @ C )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A2 @ C ) ) ) ) ).
% divides_trans
thf(fact_194_oneideal,axiom,
ideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% oneideal
thf(fact_195_zero__divides,axiom,
! [A2: a] :
( ( factor8216151070175719842xt_a_b @ r @ ( zero_a_b @ r ) @ A2 )
= ( A2
= ( zero_a_b @ r ) ) ) ).
% zero_divides
thf(fact_196_up__add__closed,axiom,
! [P: nat > a,Q: nat > a] :
( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I: nat] : ( add_a_b @ r @ ( P @ I ) @ ( Q @ I ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_add_closed
thf(fact_197_local_Ominus__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_198_add_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X2 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_199_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_200_add_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X2 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_201_add_Oinv__comm,axiom,
! [X: a,Y: a] :
( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_202_m__rcancel,axiom,
! [A2: a,B3: a,C: a] :
( ( A2
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B3 @ A2 )
= ( mult_a_ring_ext_a_b @ r @ C @ A2 ) )
= ( B3 = C ) ) ) ) ) ) ).
% m_rcancel
thf(fact_203_m__lcancel,axiom,
! [A2: a,B3: a,C: a] :
( ( A2
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A2 @ B3 )
= ( mult_a_ring_ext_a_b @ r @ A2 @ C ) )
= ( B3 = C ) ) ) ) ) ) ).
% m_lcancel
thf(fact_204_integral__iff,axiom,
! [A2: a,B3: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A2 @ B3 )
= ( zero_a_b @ r ) )
= ( ( A2
= ( zero_a_b @ r ) )
| ( B3
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_205_local_Ointegral,axiom,
! [A2: a,B3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A2 @ B3 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A2
= ( zero_a_b @ r ) )
| ( B3
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_206_r__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_207_l__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_208_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_209_inv__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% inv_unique
thf(fact_210_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_211_divides__prod__r,axiom,
! [A2: a,B3: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A2 @ B3 )
=> ( ( 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 @ B3 @ C ) ) ) ) ) ).
% divides_prod_r
thf(fact_212_divides__prod__l,axiom,
! [A2: a,B3: a,C: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A2 @ B3 )
=> ( factor8216151070175719842xt_a_b @ r @ A2 @ ( mult_a_ring_ext_a_b @ r @ C @ B3 ) ) ) ) ) ) ).
% divides_prod_l
thf(fact_213_local_Odivides__mult,axiom,
! [A2: a,C: a,B3: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A2 @ B3 )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A2 ) @ ( mult_a_ring_ext_a_b @ r @ C @ B3 ) ) ) ) ) ).
% local.divides_mult
thf(fact_214_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_215_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 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ K )
& ? [Y3: a] :
( ( member_a @ Y3 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ A2 ) @ Y3 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_216_exists__gen,axiom,
! [I3: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( I3
= ( cgenid547466209912283029xt_a_b @ r @ X2 ) ) ) ) ).
% exists_gen
thf(fact_217_cgenideal__ideal,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ A2 ) @ r ) ) ).
% cgenideal_ideal
thf(fact_218_cgenideal__minimal,axiom,
! [J2: set_a,A2: a] :
( ( ideal_a_b @ J2 @ r )
=> ( ( member_a @ A2 @ J2 )
=> ( ord_less_eq_set_a @ ( cgenid547466209912283029xt_a_b @ r @ A2 ) @ J2 ) ) ) ).
% cgenideal_minimal
thf(fact_219_genideal__minimal,axiom,
! [I3: set_a,S: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ( ( ord_less_eq_set_a @ S @ I3 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ S ) @ I3 ) ) ) ).
% genideal_minimal
thf(fact_220_genideal__ideal,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ideal_a_b @ ( genideal_a_b @ r @ S ) @ r ) ) ).
% genideal_ideal
thf(fact_221_Idl__subset__ideal,axiom,
! [I3: set_a,H2: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H2 ) @ I3 )
= ( ord_less_eq_set_a @ H2 @ I3 ) ) ) ) ).
% Idl_subset_ideal
thf(fact_222_ideal__is__subalgebra,axiom,
! [K: set_a,I3: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ideal_a_b @ I3 @ r )
=> ( embedd9027525575939734154ra_a_b @ K @ I3 @ r ) ) ) ).
% ideal_is_subalgebra
thf(fact_223_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A6: a] :
( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A6
!= ( zero_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A6 @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_224_finite__Int,axiom,
! [F2: set_nat,G2: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G2 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_225_finite__Int,axiom,
! [F2: set_a,G2: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G2 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_226_finite__Int,axiom,
! [F2: set_list_a,G2: set_list_a] :
( ( ( finite_finite_list_a @ F2 )
| ( finite_finite_list_a @ G2 ) )
=> ( finite_finite_list_a @ ( inf_inf_set_list_a @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_227_funcset__Int__left,axiom,
! [F: set_list_a > a,A: set_set_list_a,C2: set_a,B2: set_set_list_a] :
( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : C2 ) )
=> ( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ B2
@ ^ [Uu: set_list_a] : C2 ) )
=> ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ ( inf_in4657809108759609906list_a @ A @ B2 )
@ ^ [Uu: set_list_a] : C2 ) ) ) ) ).
% funcset_Int_left
thf(fact_228_funcset__Int__left,axiom,
! [F: nat > list_a,A: set_nat,C2: set_list_a,B2: set_nat] :
( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ A
@ ^ [Uu: nat] : C2 ) )
=> ( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ B2
@ ^ [Uu: nat] : C2 ) )
=> ( member_nat_list_a @ F
@ ( pi_nat_list_a @ ( inf_inf_set_nat @ A @ B2 )
@ ^ [Uu: nat] : C2 ) ) ) ) ).
% funcset_Int_left
thf(fact_229_funcset__Int__left,axiom,
! [F: nat > a,A: set_nat,C2: set_a,B2: set_nat] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : C2 ) )
=> ( ( member_nat_a @ F
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : C2 ) )
=> ( member_nat_a @ F
@ ( pi_nat_a @ ( inf_inf_set_nat @ A @ B2 )
@ ^ [Uu: nat] : C2 ) ) ) ) ).
% funcset_Int_left
thf(fact_230_funcset__Int__left,axiom,
! [F: list_a > a,A: set_list_a,C2: set_a,B2: set_list_a] :
( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : C2 ) )
=> ( ( member_list_a_a @ F
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : C2 ) )
=> ( member_list_a_a @ F
@ ( pi_list_a_a @ ( inf_inf_set_list_a @ A @ B2 )
@ ^ [Uu: list_a] : C2 ) ) ) ) ).
% funcset_Int_left
thf(fact_231_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_232_a__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_233_local_Oadd_Oright__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X )
= ( add_a_b @ r @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_234_m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_235_divides__refl,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A2 @ A2 ) ) ).
% divides_refl
thf(fact_236_r__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
= X ) ) ).
% r_zero
thf(fact_237_l__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
= X ) ) ).
% l_zero
thf(fact_238_add_Or__cancel__one_H,axiom,
! [X: a,A2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ A2 @ X ) )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_239_add_Or__cancel__one,axiom,
! [X: a,A2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A2 @ X )
= X )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_240_add_Ol__cancel__one_H,axiom,
! [X: a,A2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ X @ A2 ) )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_241_add_Ol__cancel__one,axiom,
! [X: a,A2: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ A2 )
= X )
= ( A2
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_242_r__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_243_l__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_244_r__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
= X ) ) ).
% r_one
thf(fact_245_l__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
= X ) ) ).
% l_one
thf(fact_246_divides__mult__rI,axiom,
! [A2: a,B3: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A2 @ B3 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( 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 @ B3 @ C ) ) ) ) ) ) ).
% divides_mult_rI
thf(fact_247_divides__mult__lI,axiom,
! [A2: a,B3: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A2 @ B3 )
=> ( ( 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 @ B3 ) ) ) ) ) ).
% divides_mult_lI
thf(fact_248_r__right__minus__eq,axiom,
! [A2: a,B3: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A2 @ B3 )
= ( zero_a_b @ r ) )
= ( A2 = B3 ) ) ) ) ).
% r_right_minus_eq
thf(fact_249_finprod__multf,axiom,
! [F: list_a > a,A: set_list_a,G: list_a > a] :
( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r
@ ^ [X3: list_a] : ( mult_a_ring_ext_a_b @ r @ ( F @ X3 ) @ ( G @ X3 ) )
@ A )
= ( mult_a_ring_ext_a_b @ r @ ( finpro6052973074229812797list_a @ r @ F @ A ) @ ( finpro6052973074229812797list_a @ r @ G @ A ) ) ) ) ) ).
% finprod_multf
thf(fact_250_finprod__multf,axiom,
! [F: set_list_a > a,A: set_set_list_a,G: set_list_a > a] :
( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3826550488720007709list_a @ r
@ ^ [X3: set_list_a] : ( mult_a_ring_ext_a_b @ r @ ( F @ X3 ) @ ( G @ X3 ) )
@ A )
= ( mult_a_ring_ext_a_b @ r @ ( finpro3826550488720007709list_a @ r @ F @ A ) @ ( finpro3826550488720007709list_a @ r @ G @ A ) ) ) ) ) ).
% finprod_multf
thf(fact_251_finprod__multf,axiom,
! [F: nat > a,A: set_nat,G: nat > a] :
( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r
@ ^ [X3: nat] : ( mult_a_ring_ext_a_b @ r @ ( F @ X3 ) @ ( G @ X3 ) )
@ A )
= ( mult_a_ring_ext_a_b @ r @ ( finpro1280035270526425175_b_nat @ r @ F @ A ) @ ( finpro1280035270526425175_b_nat @ r @ G @ A ) ) ) ) ) ).
% finprod_multf
thf(fact_252_finprod__multf,axiom,
! [F: a > a,A: set_a,G: a > a] :
( ( member_a_a @ F
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r
@ ^ [X3: a] : ( mult_a_ring_ext_a_b @ r @ ( F @ X3 ) @ ( G @ X3 ) )
@ A )
= ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ F @ A ) @ ( finpro205304725090349623_a_b_a @ r @ G @ A ) ) ) ) ) ).
% finprod_multf
thf(fact_253_Ring_Ointegral,axiom,
! [R: partia6043505979758434576t_unit,A2: set_a,B3: set_a] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ A2 @ B3 )
= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( member_set_a @ A2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B3 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( A2
= ( zero_s2174465271003423091t_unit @ R ) )
| ( B3
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_254_Ring_Ointegral,axiom,
! [R: partia2175431115845679010xt_a_b,A2: a,B3: a] :
( ( field_a_b @ R )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A2 @ B3 )
= ( zero_a_b @ R ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A2
= ( zero_a_b @ R ) )
| ( B3
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_255_Ring_Ointegral,axiom,
! [R: partia2670972154091845814t_unit,A2: list_a,B3: list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A2 @ B3 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( A2
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B3
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_256_Ring_Ointegral,axiom,
! [R: partia7496981018696276118t_unit,A2: set_list_a,B3: set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A2 @ B3 )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( A2
= ( zero_s2910681146719230829t_unit @ R ) )
| ( B3
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_257_Ring_Ointegral,axiom,
! [R: partia2956882679547061052t_unit,A2: list_list_a,B3: list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ A2 @ B3 )
= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( member_list_list_a @ A2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( A2
= ( zero_l347298301471573063t_unit @ R ) )
| ( B3
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_258_abelian__monoidE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_259_abelian__monoidE_I4_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_260_abelian__monoidE_I4_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( abelia3322010900105369177t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_261_abelian__monoidE_I4_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( abelia3641329199688042803t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_262_abelian__monoid_Ol__zero,axiom,
! [G2: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_monoid_a_b @ G2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( add_a_b @ G2 @ ( zero_a_b @ G2 ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_263_abelian__monoid_Ol__zero,axiom,
! [G2: partia2670972154091845814t_unit,X: list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( add_li7652885771158616974t_unit @ G2 @ ( zero_l4142658623432671053t_unit @ G2 ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_264_abelian__monoid_Ol__zero,axiom,
! [G2: partia7496981018696276118t_unit,X: set_list_a] :
( ( abelia3322010900105369177t_unit @ G2 )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( add_se2486902527185523630t_unit @ G2 @ ( zero_s2910681146719230829t_unit @ G2 ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_265_abelian__monoid_Ol__zero,axiom,
! [G2: partia2956882679547061052t_unit,X: list_list_a] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( add_li174743652000525320t_unit @ G2 @ ( zero_l347298301471573063t_unit @ G2 ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_266_abelian__monoid_Or__zero,axiom,
! [G2: partia2175431115845679010xt_a_b,X: a] :
( ( abelian_monoid_a_b @ G2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( add_a_b @ G2 @ X @ ( zero_a_b @ G2 ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_267_abelian__monoid_Or__zero,axiom,
! [G2: partia2670972154091845814t_unit,X: list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( add_li7652885771158616974t_unit @ G2 @ X @ ( zero_l4142658623432671053t_unit @ G2 ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_268_abelian__monoid_Or__zero,axiom,
! [G2: partia7496981018696276118t_unit,X: set_list_a] :
( ( abelia3322010900105369177t_unit @ G2 )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( add_se2486902527185523630t_unit @ G2 @ X @ ( zero_s2910681146719230829t_unit @ G2 ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_269_abelian__monoid_Or__zero,axiom,
! [G2: partia2956882679547061052t_unit,X: list_list_a] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( add_li174743652000525320t_unit @ G2 @ X @ ( zero_l347298301471573063t_unit @ G2 ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_270_abelian__monoid_Ominus__unique,axiom,
! [G2: partia2175431115845679010xt_a_b,Y: a,X: a,Y2: a] :
( ( abelian_monoid_a_b @ G2 )
=> ( ( ( add_a_b @ G2 @ Y @ X )
= ( zero_a_b @ G2 ) )
=> ( ( ( add_a_b @ G2 @ X @ Y2 )
= ( zero_a_b @ G2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_271_abelian__monoid_Ominus__unique,axiom,
! [G2: partia2670972154091845814t_unit,Y: list_a,X: list_a,Y2: list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( ( add_li7652885771158616974t_unit @ G2 @ Y @ X )
= ( zero_l4142658623432671053t_unit @ G2 ) )
=> ( ( ( add_li7652885771158616974t_unit @ G2 @ X @ Y2 )
= ( zero_l4142658623432671053t_unit @ G2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_272_abelian__monoid_Ominus__unique,axiom,
! [G2: partia7496981018696276118t_unit,Y: set_list_a,X: set_list_a,Y2: set_list_a] :
( ( abelia3322010900105369177t_unit @ G2 )
=> ( ( ( add_se2486902527185523630t_unit @ G2 @ Y @ X )
= ( zero_s2910681146719230829t_unit @ G2 ) )
=> ( ( ( add_se2486902527185523630t_unit @ G2 @ X @ Y2 )
= ( zero_s2910681146719230829t_unit @ G2 ) )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( member_set_list_a @ Y2 @ ( partia141011252114345353t_unit @ G2 ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_273_abelian__monoid_Ominus__unique,axiom,
! [G2: partia2956882679547061052t_unit,Y: list_list_a,X: list_list_a,Y2: list_list_a] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( ( add_li174743652000525320t_unit @ G2 @ Y @ X )
= ( zero_l347298301471573063t_unit @ G2 ) )
=> ( ( ( add_li174743652000525320t_unit @ G2 @ X @ Y2 )
= ( zero_l347298301471573063t_unit @ G2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_274_abelian__monoidI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X2: a,Y4: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X2 @ Y4 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ! [X2: a,Y4: a,Z2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X2 @ Y4 ) @ Z2 )
= ( add_a_b @ R @ X2 @ ( add_a_b @ R @ Y4 @ Z2 ) ) ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X2 )
= X2 ) )
=> ( ! [X2: a,Y4: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X2 @ Y4 )
= ( add_a_b @ R @ Y4 @ X2 ) ) ) )
=> ( abelian_monoid_a_b @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_275_abelian__monoidI,axiom,
! [R: partia2670972154091845814t_unit] :
( ! [X2: list_a,Y4: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y4 ) @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ! [X2: list_a,Y4: list_a,Z2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y4 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ X2 @ ( add_li7652885771158616974t_unit @ R @ Y4 @ Z2 ) ) ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X2 )
= X2 ) )
=> ( ! [X2: list_a,Y4: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X2 @ Y4 )
= ( add_li7652885771158616974t_unit @ R @ Y4 @ X2 ) ) ) )
=> ( abelia226231641709521465t_unit @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_276_abelian__monoidI,axiom,
! [R: partia7496981018696276118t_unit] :
( ! [X2: set_list_a,Y4: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ R ) )
=> ( member_set_list_a @ ( add_se2486902527185523630t_unit @ R @ X2 @ Y4 ) @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ! [X2: set_list_a,Y4: set_list_a,Z2: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Z2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( add_se2486902527185523630t_unit @ R @ X2 @ Y4 ) @ Z2 )
= ( add_se2486902527185523630t_unit @ R @ X2 @ ( add_se2486902527185523630t_unit @ R @ Y4 @ Z2 ) ) ) ) ) )
=> ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X2 )
= X2 ) )
=> ( ! [X2: set_list_a,Y4: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X2 @ Y4 )
= ( add_se2486902527185523630t_unit @ R @ Y4 @ X2 ) ) ) )
=> ( abelia3322010900105369177t_unit @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_277_abelian__monoidI,axiom,
! [R: partia2956882679547061052t_unit] :
( ! [X2: list_list_a,Y4: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X2 @ Y4 ) @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ! [X2: list_list_a,Y4: list_list_a,Z2: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X2 @ Y4 ) @ Z2 )
= ( add_li174743652000525320t_unit @ R @ X2 @ ( add_li174743652000525320t_unit @ R @ Y4 @ Z2 ) ) ) ) ) )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X2 )
= X2 ) )
=> ( ! [X2: list_list_a,Y4: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X2 @ Y4 )
= ( add_li174743652000525320t_unit @ R @ Y4 @ X2 ) ) ) )
=> ( abelia3641329199688042803t_unit @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_278_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_279_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_280_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X @ ( zero_s2910681146719230829t_unit @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_281_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X @ ( zero_l347298301471573063t_unit @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_282_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_283_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_284_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_285_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_286_semiring_Ol__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_287_semiring_Ol__distr,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_288_semiring_Ol__distr,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ ( add_se2486902527185523630t_unit @ R @ X @ Y ) @ Z )
= ( add_se2486902527185523630t_unit @ R @ ( mult_s7802724872828879953t_unit @ R @ X @ Z ) @ ( mult_s7802724872828879953t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_289_semiring_Ol__distr,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Z ) @ ( mult_l4853965630390486993t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_290_semiring_Or__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_291_semiring_Or__distr,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( 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 @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_292_semiring_Or__distr,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ Z @ ( add_se2486902527185523630t_unit @ R @ X @ Y ) )
= ( add_se2486902527185523630t_unit @ R @ ( mult_s7802724872828879953t_unit @ R @ Z @ X ) @ ( mult_s7802724872828879953t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_293_semiring_Or__distr,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( 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 @ X @ Y ) )
= ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ Z @ X ) @ ( mult_l4853965630390486993t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_294_semiring_Ol__null,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.l_null
thf(fact_295_semiring_Ol__null,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.l_null
thf(fact_296_semiring_Ol__null,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% semiring.l_null
thf(fact_297_semiring_Ol__null,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.l_null
thf(fact_298_semiring_Or__null,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.r_null
thf(fact_299_semiring_Or__null,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.r_null
thf(fact_300_semiring_Or__null,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ X @ ( zero_s2910681146719230829t_unit @ R ) )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% semiring.r_null
thf(fact_301_semiring_Or__null,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ X @ ( zero_l347298301471573063t_unit @ R ) )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.r_null
thf(fact_302_abelian__monoidE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y )
= ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_303_abelian__monoidE_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_304_abelian__monoidE_I5_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
( ( abelia3322010900105369177t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X @ Y )
= ( add_se2486902527185523630t_unit @ R @ Y @ X ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_305_abelian__monoidE_I5_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( abelia3641329199688042803t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X @ Y )
= ( add_li174743652000525320t_unit @ R @ Y @ X ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_306_abelian__monoidE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_307_abelian__monoidE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_308_abelian__monoidE_I3_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( abelia3322010900105369177t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( add_se2486902527185523630t_unit @ R @ X @ Y ) @ Z )
= ( add_se2486902527185523630t_unit @ R @ X @ ( add_se2486902527185523630t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_309_abelian__monoidE_I3_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( abelia3641329199688042803t_unit @ R )
=> ( ( member_list_list_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_310_abelian__monoidE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_311_abelian__monoidE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_312_abelian__monoidE_I1_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
( ( abelia3322010900105369177t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
=> ( member_set_list_a @ ( add_se2486902527185523630t_unit @ R @ X @ Y ) @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_313_abelian__monoidE_I1_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( abelia3641329199688042803t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_314_abelian__monoid_Oa__comm,axiom,
! [G2: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ G2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( add_a_b @ G2 @ X @ Y )
= ( add_a_b @ G2 @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_315_abelian__monoid_Oa__comm,axiom,
! [G2: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( add_li7652885771158616974t_unit @ G2 @ X @ Y )
= ( add_li7652885771158616974t_unit @ G2 @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_316_abelian__monoid_Oa__comm,axiom,
! [G2: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
( ( abelia3322010900105369177t_unit @ G2 )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( add_se2486902527185523630t_unit @ G2 @ X @ Y )
= ( add_se2486902527185523630t_unit @ G2 @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_317_abelian__monoid_Oa__comm,axiom,
! [G2: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( add_li174743652000525320t_unit @ G2 @ X @ Y )
= ( add_li174743652000525320t_unit @ G2 @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_318_abelian__monoid_Oa__assoc,axiom,
! [G2: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ G2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( add_a_b @ G2 @ ( add_a_b @ G2 @ X @ Y ) @ Z )
= ( add_a_b @ G2 @ X @ ( add_a_b @ G2 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_319_abelian__monoid_Oa__assoc,axiom,
! [G2: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( add_li7652885771158616974t_unit @ G2 @ ( add_li7652885771158616974t_unit @ G2 @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ G2 @ X @ ( add_li7652885771158616974t_unit @ G2 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_320_abelian__monoid_Oa__assoc,axiom,
! [G2: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( abelia3322010900105369177t_unit @ G2 )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( add_se2486902527185523630t_unit @ G2 @ ( add_se2486902527185523630t_unit @ G2 @ X @ Y ) @ Z )
= ( add_se2486902527185523630t_unit @ G2 @ X @ ( add_se2486902527185523630t_unit @ G2 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_321_abelian__monoid_Oa__assoc,axiom,
! [G2: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( add_li174743652000525320t_unit @ G2 @ ( add_li174743652000525320t_unit @ G2 @ X @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ G2 @ X @ ( add_li174743652000525320t_unit @ G2 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_322_abelian__monoid_Oa__lcomm,axiom,
! [G2: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( abelian_monoid_a_b @ G2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( add_a_b @ G2 @ X @ ( add_a_b @ G2 @ Y @ Z ) )
= ( add_a_b @ G2 @ Y @ ( add_a_b @ G2 @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_323_abelian__monoid_Oa__lcomm,axiom,
! [G2: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( add_li7652885771158616974t_unit @ G2 @ X @ ( add_li7652885771158616974t_unit @ G2 @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ G2 @ Y @ ( add_li7652885771158616974t_unit @ G2 @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_324_abelian__monoid_Oa__lcomm,axiom,
! [G2: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( abelia3322010900105369177t_unit @ G2 )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( add_se2486902527185523630t_unit @ G2 @ X @ ( add_se2486902527185523630t_unit @ G2 @ Y @ Z ) )
= ( add_se2486902527185523630t_unit @ G2 @ Y @ ( add_se2486902527185523630t_unit @ G2 @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_325_abelian__monoid_Oa__lcomm,axiom,
! [G2: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( add_li174743652000525320t_unit @ G2 @ X @ ( add_li174743652000525320t_unit @ G2 @ Y @ Z ) )
= ( add_li174743652000525320t_unit @ G2 @ Y @ ( add_li174743652000525320t_unit @ G2 @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_326_abelian__monoid_Oa__closed,axiom,
! [G2: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( abelian_monoid_a_b @ G2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( member_a @ ( add_a_b @ G2 @ X @ Y ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_327_abelian__monoid_Oa__closed,axiom,
! [G2: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ G2 @ X @ Y ) @ ( partia5361259788508890537t_unit @ G2 ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_328_abelian__monoid_Oa__closed,axiom,
! [G2: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
( ( abelia3322010900105369177t_unit @ G2 )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ G2 ) )
=> ( member_set_list_a @ ( add_se2486902527185523630t_unit @ G2 @ X @ Y ) @ ( partia141011252114345353t_unit @ G2 ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_329_abelian__monoid_Oa__closed,axiom,
! [G2: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ G2 @ X @ Y ) @ ( partia2464479390973590831t_unit @ G2 ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_330_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_331_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_332_abelian__monoidE_I2_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( abelia3322010900105369177t_unit @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_333_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_334_abelian__monoid_Ozero__closed,axiom,
! [G2: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ G2 )
=> ( member_a @ ( zero_a_b @ G2 ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_335_abelian__monoid_Ozero__closed,axiom,
! [G2: partia2670972154091845814t_unit] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ G2 ) @ ( partia5361259788508890537t_unit @ G2 ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_336_abelian__monoid_Ozero__closed,axiom,
! [G2: partia7496981018696276118t_unit] :
( ( abelia3322010900105369177t_unit @ G2 )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ G2 ) @ ( partia141011252114345353t_unit @ G2 ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_337_abelian__monoid_Ozero__closed,axiom,
! [G2: partia2956882679547061052t_unit] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ G2 ) @ ( partia2464479390973590831t_unit @ G2 ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_338_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_339_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ R @ Y @ ( add_li7652885771158616974t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_340_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X @ ( add_se2486902527185523630t_unit @ R @ Y @ Z ) )
= ( add_se2486902527185523630t_unit @ R @ Y @ ( add_se2486902527185523630t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_341_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) )
= ( add_li174743652000525320t_unit @ R @ Y @ ( add_li174743652000525320t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_342_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y )
= ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_343_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_344_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X @ Y )
= ( add_se2486902527185523630t_unit @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_345_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X @ Y )
= ( add_li174743652000525320t_unit @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_346_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_347_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_348_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( add_se2486902527185523630t_unit @ R @ X @ Y ) @ Z )
= ( add_se2486902527185523630t_unit @ R @ X @ ( add_se2486902527185523630t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_349_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_350_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_351_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_352_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
=> ( member_set_list_a @ ( add_se2486902527185523630t_unit @ R @ X @ Y ) @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_353_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_354_Ring_Oone__not__zero,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( one_se211549098623999037t_unit @ R )
!= ( zero_s2174465271003423091t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_355_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_356_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_357_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_358_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_359_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_360_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_361_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_362_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ R @ X @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_363_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ ( mult_s7802724872828879953t_unit @ R @ X @ Y ) @ Z )
= ( mult_s7802724872828879953t_unit @ R @ X @ ( mult_s7802724872828879953t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_364_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( 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 @ X @ Y ) @ Z )
= ( mult_l4853965630390486993t_unit @ R @ X @ ( mult_l4853965630390486993t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_365_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_366_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_367_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
=> ( member_set_list_a @ ( mult_s7802724872828879953t_unit @ R @ X @ Y ) @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_368_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_369_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_370_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_371_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ ( one_se1127990129394575805t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_372_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( one_li8234411390022467901t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_373_semiring_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( abelian_monoid_a_b @ R ) ) ).
% semiring.axioms(1)
thf(fact_374_semiring_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( abelia226231641709521465t_unit @ R ) ) ).
% semiring.axioms(1)
thf(fact_375_exp__base__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% exp_base_closed
thf(fact_376_isgcd__divides__r,axiom,
! [B3: a,A2: a] :
( ( factor8216151070175719842xt_a_b @ r @ B3 @ A2 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ B3 @ A2 @ B3 ) ) ) ) ).
% isgcd_divides_r
thf(fact_377_isgcd__divides__l,axiom,
! [A2: a,B3: a] :
( ( factor8216151070175719842xt_a_b @ r @ A2 @ B3 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ A2 @ A2 @ B3 ) ) ) ) ).
% isgcd_divides_l
thf(fact_378_bound__upD,axiom,
! [F: nat > a] :
( ( member_nat_a @ F @ ( up_a_b @ r ) )
=> ? [N2: nat] : ( bound_a @ ( zero_a_b @ r ) @ N2 @ F ) ) ).
% bound_upD
thf(fact_379_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_380_ring__iso__imp__img__field,axiom,
! [H: a > a,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ r @ S ) )
=> ( field_a_b
@ ( zero_update_a_b
@ ^ [Uu: a] : ( H @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_field
thf(fact_381_ring__iso__imp__img__field,axiom,
! [H: a > list_a,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ r @ S ) )
=> ( field_6388047844668329575t_unit
@ ( zero_u1196785550890449590t_unit
@ ^ [Uu: list_a] : ( H @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_field
thf(fact_382_ring__iso__imp__img__field,axiom,
! [H: a > set_a,S: partia6043505979758434576t_unit] :
( ( member_a_set_a @ H @ ( ring_i7849008455817099456t_unit @ r @ S ) )
=> ( field_6045675692312731021t_unit
@ ( zero_u8960205505688764764t_unit
@ ^ [Uu: set_a] : ( H @ ( zero_a_b @ r ) )
@ S ) ) ) ).
% ring_iso_imp_img_field
thf(fact_383_ring__irreducibleE_I1_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( R2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_384_dividesI_H,axiom,
! [B3: a,G2: partia2175431115845679010xt_a_b,A2: a,C: a] :
( ( B3
= ( mult_a_ring_ext_a_b @ G2 @ A2 @ C ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( factor8216151070175719842xt_a_b @ G2 @ A2 @ B3 ) ) ) ).
% dividesI'
thf(fact_385_dividesI_H,axiom,
! [B3: list_a,G2: partia2670972154091845814t_unit,A2: list_a,C: list_a] :
( ( B3
= ( mult_l7073676228092353617t_unit @ G2 @ A2 @ C ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( factor1757716651909850160t_unit @ G2 @ A2 @ B3 ) ) ) ).
% dividesI'
thf(fact_386_dividesI_H,axiom,
! [B3: set_list_a,G2: partia7496981018696276118t_unit,A2: set_list_a,C: set_list_a] :
( ( B3
= ( mult_s7802724872828879953t_unit @ G2 @ A2 @ C ) )
=> ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ G2 ) )
=> ( factor2800830226752492592t_unit @ G2 @ A2 @ B3 ) ) ) ).
% dividesI'
thf(fact_387_dividesI_H,axiom,
! [B3: set_list_a,G2: partia1167524930811108609t_unit,A2: set_list_a,C: set_list_a] :
( ( B3
= ( mult_s5375104596032733786t_unit @ G2 @ A2 @ C ) )
=> ( ( member_set_list_a @ C @ ( partia5178357399839081912t_unit @ G2 ) )
=> ( factor4555674416838057147t_unit @ G2 @ A2 @ B3 ) ) ) ).
% dividesI'
thf(fact_388_dividesI_H,axiom,
! [B3: list_list_a,G2: partia2956882679547061052t_unit,A2: list_list_a,C: list_list_a] :
( ( B3
= ( mult_l4853965630390486993t_unit @ G2 @ A2 @ C ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( factor6954119973539764400t_unit @ G2 @ A2 @ B3 ) ) ) ).
% dividesI'
thf(fact_389_noetherian__ring_Ofinetely__gen,axiom,
! [R: partia7496981018696276118t_unit,I3: set_set_list_a] :
( ( ring_n7704429503468267069t_unit @ R )
=> ( ( ideal_5294671857925479125t_unit @ I3 @ R )
=> ? [A5: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A5 @ ( partia141011252114345353t_unit @ R ) )
& ( finite5282473924520328461list_a @ A5 )
& ( I3
= ( genide4187322989772540535t_unit @ R @ A5 ) ) ) ) ) ).
% noetherian_ring.finetely_gen
thf(fact_390_noetherian__ring_Ofinetely__gen,axiom,
! [R: partia2956882679547061052t_unit,I3: set_list_list_a] :
( ( ring_n1719824158142654231t_unit @ R )
=> ( ( ideal_7391923968229085103t_unit @ I3 @ R )
=> ? [A5: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ A5 @ ( partia2464479390973590831t_unit @ R ) )
& ( finite1660835950917165235list_a @ A5 )
& ( I3
= ( genide2671672708880404049t_unit @ R @ A5 ) ) ) ) ) ).
% noetherian_ring.finetely_gen
thf(fact_391_noetherian__ring_Ofinetely__gen,axiom,
! [R: partia2175431115845679010xt_a_b,I3: set_a] :
( ( ring_n3639167112692572309ng_a_b @ R )
=> ( ( ideal_a_b @ I3 @ R )
=> ? [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ ( partia707051561876973205xt_a_b @ R ) )
& ( finite_finite_a @ A5 )
& ( I3
= ( genideal_a_b @ R @ A5 ) ) ) ) ) ).
% noetherian_ring.finetely_gen
thf(fact_392_noetherian__ring_Ofinetely__gen,axiom,
! [R: partia2670972154091845814t_unit,I3: set_list_a] :
( ( ring_n5188127996776581661t_unit @ R )
=> ( ( ideal_8896367198367571637t_unit @ I3 @ R )
=> ? [A5: set_list_a] :
( ( ord_le8861187494160871172list_a @ A5 @ ( partia5361259788508890537t_unit @ R ) )
& ( finite_finite_list_a @ A5 )
& ( I3
= ( genide3243992037924705879t_unit @ R @ A5 ) ) ) ) ) ).
% noetherian_ring.finetely_gen
thf(fact_393_finprod__Un__Int,axiom,
! [A: set_set_list_a,B2: set_set_list_a,G: set_list_a > a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( finite5282473924520328461list_a @ B2 )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ B2
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( finpro3826550488720007709list_a @ r @ G @ ( sup_su4537662296134749976list_a @ A @ B2 ) ) @ ( finpro3826550488720007709list_a @ r @ G @ ( inf_in4657809108759609906list_a @ A @ B2 ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro3826550488720007709list_a @ r @ G @ A ) @ ( finpro3826550488720007709list_a @ r @ G @ B2 ) ) ) ) ) ) ) ).
% finprod_Un_Int
thf(fact_394_finprod__Un__Int,axiom,
! [A: set_nat,B2: set_nat,G: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( finpro1280035270526425175_b_nat @ r @ G @ ( sup_sup_set_nat @ A @ B2 ) ) @ ( finpro1280035270526425175_b_nat @ r @ G @ ( inf_inf_set_nat @ A @ B2 ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro1280035270526425175_b_nat @ r @ G @ A ) @ ( finpro1280035270526425175_b_nat @ r @ G @ B2 ) ) ) ) ) ) ) ).
% finprod_Un_Int
thf(fact_395_finprod__Un__Int,axiom,
! [A: set_list_a,B2: set_list_a,G: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( finpro6052973074229812797list_a @ r @ G @ ( sup_sup_set_list_a @ A @ B2 ) ) @ ( finpro6052973074229812797list_a @ r @ G @ ( inf_inf_set_list_a @ A @ B2 ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro6052973074229812797list_a @ r @ G @ A ) @ ( finpro6052973074229812797list_a @ r @ G @ B2 ) ) ) ) ) ) ) ).
% finprod_Un_Int
thf(fact_396_finprod__Un__Int,axiom,
! [A: set_a,B2: set_a,G: a > a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ G @ ( sup_sup_set_a @ A @ B2 ) ) @ ( finpro205304725090349623_a_b_a @ r @ G @ ( inf_inf_set_a @ A @ B2 ) ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ G @ A ) @ ( finpro205304725090349623_a_b_a @ r @ G @ B2 ) ) ) ) ) ) ) ).
% finprod_Un_Int
thf(fact_397_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_398_finite__Un,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_399_finite__Un,axiom,
! [F2: set_list_a,G2: set_list_a] :
( ( finite_finite_list_a @ ( sup_sup_set_list_a @ F2 @ G2 ) )
= ( ( finite_finite_list_a @ F2 )
& ( finite_finite_list_a @ G2 ) ) ) ).
% finite_Un
thf(fact_400_finite__Un,axiom,
! [F2: set_a,G2: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G2 ) )
= ( ( finite_finite_a @ F2 )
& ( finite_finite_a @ G2 ) ) ) ).
% finite_Un
thf(fact_401_funcset__Un__left,axiom,
! [F: set_list_a > a,A: set_set_list_a,B2: set_set_list_a,C2: set_a] :
( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ ( sup_su4537662296134749976list_a @ A @ B2 )
@ ^ [Uu: set_list_a] : C2 ) )
= ( ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : C2 ) )
& ( member_set_list_a_a @ F
@ ( pi_set_list_a_a @ B2
@ ^ [Uu: set_list_a] : C2 ) ) ) ) ).
% funcset_Un_left
thf(fact_402_funcset__Un__left,axiom,
! [F: nat > list_a,A: set_nat,B2: set_nat,C2: set_list_a] :
( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ ( sup_sup_set_nat @ A @ B2 )
@ ^ [Uu: nat] : C2 ) )
= ( ( member_nat_list_a @ F
@ ( pi_nat_list_a @ A
@ ^ [Uu: nat] : C2 ) )
& ( member_nat_list_a @ F
@ ( pi_nat_list_a @ B2
@ ^ [Uu: nat] : C2 ) ) ) ) ).
% funcset_Un_left
thf(fact_403_funcset__Un__left,axiom,
! [F: nat > a,A: set_nat,B2: set_nat,C2: set_a] :
( ( member_nat_a @ F
@ ( pi_nat_a @ ( sup_sup_set_nat @ A @ B2 )
@ ^ [Uu: nat] : C2 ) )
= ( ( member_nat_a @ F
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : C2 ) )
& ( member_nat_a @ F
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : C2 ) ) ) ) ).
% funcset_Un_left
thf(fact_404_funcset__Un__left,axiom,
! [F: list_a > a,A: set_list_a,B2: set_list_a,C2: set_a] :
( ( member_list_a_a @ F
@ ( pi_list_a_a @ ( sup_sup_set_list_a @ A @ B2 )
@ ^ [Uu: list_a] : C2 ) )
= ( ( member_list_a_a @ F
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : C2 ) )
& ( member_list_a_a @ F
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : C2 ) ) ) ) ).
% funcset_Un_left
thf(fact_405_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_406_infinite__Un,axiom,
! [S: set_list_a,T: set_list_a] :
( ( ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S @ T ) ) )
= ( ~ ( finite_finite_list_a @ S )
| ~ ( finite_finite_list_a @ T ) ) ) ).
% infinite_Un
thf(fact_407_infinite__Un,axiom,
! [S: set_a,T: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) )
= ( ~ ( finite_finite_a @ S )
| ~ ( finite_finite_a @ T ) ) ) ).
% infinite_Un
thf(fact_408_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_409_Un__infinite,axiom,
! [S: set_list_a,T: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_410_Un__infinite,axiom,
! [S: set_a,T: set_a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T ) ) ) ).
% Un_infinite
thf(fact_411_finite__UnI,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_412_finite__UnI,axiom,
! [F2: set_list_a,G2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ( finite_finite_list_a @ G2 )
=> ( finite_finite_list_a @ ( sup_sup_set_list_a @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_413_finite__UnI,axiom,
! [F2: set_a,G2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( finite_finite_a @ G2 )
=> ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_414_principal__domain_Oprimeness__condition,axiom,
! [R: partia7496981018696276118t_unit,P: set_list_a] :
( ( ring_p2468016639901664456t_unit @ R )
=> ( ( member_set_list_a @ P @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r5115406448772830318t_unit @ R @ P )
= ( ring_r1091214237498979717t_unit @ R @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_415_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_416_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_417_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_418_ring_Ounfold__congs_I4_J,axiom,
! [R2: partia2175431115845679010xt_a_b,R3: partia2175431115845679010xt_a_b,V3: a,F: a > a,F3: a > a] :
( ( R2 = R3 )
=> ( ( ( zero_a_b @ R3 )
= V3 )
=> ( ! [V4: a] :
( ( V4 = V3 )
=> ( ( F @ V4 )
= ( F3 @ V4 ) ) )
=> ( ( zero_update_a_b @ F @ R2 )
= ( zero_update_a_b @ F3 @ R3 ) ) ) ) ) ).
% ring.unfold_congs(4)
thf(fact_419_ring_Ounfold__congs_I4_J,axiom,
! [R2: partia2670972154091845814t_unit,R3: partia2670972154091845814t_unit,V3: list_a,F: list_a > list_a,F3: list_a > list_a] :
( ( R2 = R3 )
=> ( ( ( zero_l4142658623432671053t_unit @ R3 )
= V3 )
=> ( ! [V4: list_a] :
( ( V4 = V3 )
=> ( ( F @ V4 )
= ( F3 @ V4 ) ) )
=> ( ( zero_u1196785550890449590t_unit @ F @ R2 )
= ( zero_u1196785550890449590t_unit @ F3 @ R3 ) ) ) ) ) ).
% ring.unfold_congs(4)
thf(fact_420_ring_Ofold__congs_I4_J,axiom,
! [R2: partia2175431115845679010xt_a_b,R3: partia2175431115845679010xt_a_b,V3: a,F: a > a,F3: a > a] :
( ( R2 = R3 )
=> ( ( ( zero_a_b @ R3 )
= V3 )
=> ( ! [V4: a] :
( ( V3 = V4 )
=> ( ( F @ V4 )
= ( F3 @ V4 ) ) )
=> ( ( zero_update_a_b @ F @ R2 )
= ( zero_update_a_b @ F3 @ R3 ) ) ) ) ) ).
% ring.fold_congs(4)
thf(fact_421_ring_Ofold__congs_I4_J,axiom,
! [R2: partia2670972154091845814t_unit,R3: partia2670972154091845814t_unit,V3: list_a,F: list_a > list_a,F3: list_a > list_a] :
( ( R2 = R3 )
=> ( ( ( zero_l4142658623432671053t_unit @ R3 )
= V3 )
=> ( ! [V4: list_a] :
( ( V3 = V4 )
=> ( ( F @ V4 )
= ( F3 @ V4 ) ) )
=> ( ( zero_u1196785550890449590t_unit @ F @ R2 )
= ( zero_u1196785550890449590t_unit @ F3 @ R3 ) ) ) ) ) ).
% ring.fold_congs(4)
thf(fact_422_isgcd__def,axiom,
( isgcd_a_ring_ext_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,X3: a,A3: a,B4: a] :
( ( factor8216151070175719842xt_a_b @ G3 @ X3 @ A3 )
& ( factor8216151070175719842xt_a_b @ G3 @ X3 @ B4 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G3 ) )
=> ( ( ( factor8216151070175719842xt_a_b @ G3 @ Y3 @ A3 )
& ( factor8216151070175719842xt_a_b @ G3 @ Y3 @ B4 ) )
=> ( factor8216151070175719842xt_a_b @ G3 @ Y3 @ X3 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_423_isgcd__def,axiom,
( isgcd_1118609098697428327t_unit
= ( ^ [G3: partia2670972154091845814t_unit,X3: list_a,A3: list_a,B4: list_a] :
( ( factor1757716651909850160t_unit @ G3 @ X3 @ A3 )
& ( factor1757716651909850160t_unit @ G3 @ X3 @ B4 )
& ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G3 ) )
=> ( ( ( factor1757716651909850160t_unit @ G3 @ Y3 @ A3 )
& ( factor1757716651909850160t_unit @ G3 @ Y3 @ B4 ) )
=> ( factor1757716651909850160t_unit @ G3 @ Y3 @ X3 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_424_isgcd__def,axiom,
( isgcd_3372036304480159079t_unit
= ( ^ [G3: partia7496981018696276118t_unit,X3: set_list_a,A3: set_list_a,B4: set_list_a] :
( ( factor2800830226752492592t_unit @ G3 @ X3 @ A3 )
& ( factor2800830226752492592t_unit @ G3 @ X3 @ B4 )
& ! [Y3: set_list_a] :
( ( member_set_list_a @ Y3 @ ( partia141011252114345353t_unit @ G3 ) )
=> ( ( ( factor2800830226752492592t_unit @ G3 @ Y3 @ A3 )
& ( factor2800830226752492592t_unit @ G3 @ Y3 @ B4 ) )
=> ( factor2800830226752492592t_unit @ G3 @ Y3 @ X3 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_425_isgcd__def,axiom,
( isgcd_7987020228990422980t_unit
= ( ^ [G3: partia1167524930811108609t_unit,X3: set_list_a,A3: set_list_a,B4: set_list_a] :
( ( factor4555674416838057147t_unit @ G3 @ X3 @ A3 )
& ( factor4555674416838057147t_unit @ G3 @ X3 @ B4 )
& ! [Y3: set_list_a] :
( ( member_set_list_a @ Y3 @ ( partia5178357399839081912t_unit @ G3 ) )
=> ( ( ( factor4555674416838057147t_unit @ G3 @ Y3 @ A3 )
& ( factor4555674416838057147t_unit @ G3 @ Y3 @ B4 ) )
=> ( factor4555674416838057147t_unit @ G3 @ Y3 @ X3 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_426_isgcd__def,axiom,
( isgcd_3804025100609598183t_unit
= ( ^ [G3: partia2956882679547061052t_unit,X3: list_list_a,A3: list_list_a,B4: list_list_a] :
( ( factor6954119973539764400t_unit @ G3 @ X3 @ A3 )
& ( factor6954119973539764400t_unit @ G3 @ X3 @ B4 )
& ! [Y3: list_list_a] :
( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ G3 ) )
=> ( ( ( factor6954119973539764400t_unit @ G3 @ Y3 @ A3 )
& ( factor6954119973539764400t_unit @ G3 @ Y3 @ B4 ) )
=> ( factor6954119973539764400t_unit @ G3 @ Y3 @ X3 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_427_noetherian__domain_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_n4705423059119889713t_unit @ R )
=> ( ring_n5188127996776581661t_unit @ R ) ) ).
% noetherian_domain.axioms(1)
thf(fact_428_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_429_dividesD,axiom,
! [G2: partia2175431115845679010xt_a_b,A2: a,B3: a] :
( ( factor8216151070175719842xt_a_b @ G2 @ A2 @ B3 )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G2 ) )
& ( B3
= ( mult_a_ring_ext_a_b @ G2 @ A2 @ X2 ) ) ) ) ).
% dividesD
thf(fact_430_dividesD,axiom,
! [G2: partia2670972154091845814t_unit,A2: list_a,B3: list_a] :
( ( factor1757716651909850160t_unit @ G2 @ A2 @ B3 )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G2 ) )
& ( B3
= ( mult_l7073676228092353617t_unit @ G2 @ A2 @ X2 ) ) ) ) ).
% dividesD
thf(fact_431_dividesD,axiom,
! [G2: partia7496981018696276118t_unit,A2: set_list_a,B3: set_list_a] :
( ( factor2800830226752492592t_unit @ G2 @ A2 @ B3 )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ G2 ) )
& ( B3
= ( mult_s7802724872828879953t_unit @ G2 @ A2 @ X2 ) ) ) ) ).
% dividesD
thf(fact_432_dividesD,axiom,
! [G2: partia1167524930811108609t_unit,A2: set_list_a,B3: set_list_a] :
( ( factor4555674416838057147t_unit @ G2 @ A2 @ B3 )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia5178357399839081912t_unit @ G2 ) )
& ( B3
= ( mult_s5375104596032733786t_unit @ G2 @ A2 @ X2 ) ) ) ) ).
% dividesD
thf(fact_433_dividesD,axiom,
! [G2: partia2956882679547061052t_unit,A2: list_list_a,B3: list_list_a] :
( ( factor6954119973539764400t_unit @ G2 @ A2 @ B3 )
=> ? [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G2 ) )
& ( B3
= ( mult_l4853965630390486993t_unit @ G2 @ A2 @ X2 ) ) ) ) ).
% dividesD
thf(fact_434_dividesE,axiom,
! [G2: partia2175431115845679010xt_a_b,A2: a,B3: a] :
( ( factor8216151070175719842xt_a_b @ G2 @ A2 @ B3 )
=> ~ ! [C3: a] :
( ( B3
= ( mult_a_ring_ext_a_b @ G2 @ A2 @ C3 ) )
=> ~ ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ).
% dividesE
thf(fact_435_dividesE,axiom,
! [G2: partia2670972154091845814t_unit,A2: list_a,B3: list_a] :
( ( factor1757716651909850160t_unit @ G2 @ A2 @ B3 )
=> ~ ! [C3: list_a] :
( ( B3
= ( mult_l7073676228092353617t_unit @ G2 @ A2 @ C3 ) )
=> ~ ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ G2 ) ) ) ) ).
% dividesE
thf(fact_436_dividesE,axiom,
! [G2: partia7496981018696276118t_unit,A2: set_list_a,B3: set_list_a] :
( ( factor2800830226752492592t_unit @ G2 @ A2 @ B3 )
=> ~ ! [C3: set_list_a] :
( ( B3
= ( mult_s7802724872828879953t_unit @ G2 @ A2 @ C3 ) )
=> ~ ( member_set_list_a @ C3 @ ( partia141011252114345353t_unit @ G2 ) ) ) ) ).
% dividesE
thf(fact_437_dividesE,axiom,
! [G2: partia1167524930811108609t_unit,A2: set_list_a,B3: set_list_a] :
( ( factor4555674416838057147t_unit @ G2 @ A2 @ B3 )
=> ~ ! [C3: set_list_a] :
( ( B3
= ( mult_s5375104596032733786t_unit @ G2 @ A2 @ C3 ) )
=> ~ ( member_set_list_a @ C3 @ ( partia5178357399839081912t_unit @ G2 ) ) ) ) ).
% dividesE
thf(fact_438_dividesE,axiom,
! [G2: partia2956882679547061052t_unit,A2: list_list_a,B3: list_list_a] :
( ( factor6954119973539764400t_unit @ G2 @ A2 @ B3 )
=> ~ ! [C3: list_list_a] :
( ( B3
= ( mult_l4853965630390486993t_unit @ G2 @ A2 @ C3 ) )
=> ~ ( member_list_list_a @ C3 @ ( partia2464479390973590831t_unit @ G2 ) ) ) ) ).
% dividesE
thf(fact_439_dividesI,axiom,
! [C: a,G2: partia2175431115845679010xt_a_b,B3: a,A2: a] :
( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( B3
= ( mult_a_ring_ext_a_b @ G2 @ A2 @ C ) )
=> ( factor8216151070175719842xt_a_b @ G2 @ A2 @ B3 ) ) ) ).
% dividesI
thf(fact_440_dividesI,axiom,
! [C: list_a,G2: partia2670972154091845814t_unit,B3: list_a,A2: list_a] :
( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( B3
= ( mult_l7073676228092353617t_unit @ G2 @ A2 @ C ) )
=> ( factor1757716651909850160t_unit @ G2 @ A2 @ B3 ) ) ) ).
% dividesI
thf(fact_441_dividesI,axiom,
! [C: set_list_a,G2: partia7496981018696276118t_unit,B3: set_list_a,A2: set_list_a] :
( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( B3
= ( mult_s7802724872828879953t_unit @ G2 @ A2 @ C ) )
=> ( factor2800830226752492592t_unit @ G2 @ A2 @ B3 ) ) ) ).
% dividesI
thf(fact_442_dividesI,axiom,
! [C: set_list_a,G2: partia1167524930811108609t_unit,B3: set_list_a,A2: set_list_a] :
( ( member_set_list_a @ C @ ( partia5178357399839081912t_unit @ G2 ) )
=> ( ( B3
= ( mult_s5375104596032733786t_unit @ G2 @ A2 @ C ) )
=> ( factor4555674416838057147t_unit @ G2 @ A2 @ B3 ) ) ) ).
% dividesI
thf(fact_443_dividesI,axiom,
! [C: list_list_a,G2: partia2956882679547061052t_unit,B3: list_list_a,A2: list_list_a] :
( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( B3
= ( mult_l4853965630390486993t_unit @ G2 @ A2 @ C ) )
=> ( factor6954119973539764400t_unit @ G2 @ A2 @ B3 ) ) ) ).
% dividesI
thf(fact_444_factor__def,axiom,
( factor8216151070175719842xt_a_b
= ( ^ [G3: partia2175431115845679010xt_a_b,A3: a,B4: a] :
? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G3 ) )
& ( B4
= ( mult_a_ring_ext_a_b @ G3 @ A3 @ X3 ) ) ) ) ) ).
% factor_def
thf(fact_445_factor__def,axiom,
( factor1757716651909850160t_unit
= ( ^ [G3: partia2670972154091845814t_unit,A3: list_a,B4: list_a] :
? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G3 ) )
& ( B4
= ( mult_l7073676228092353617t_unit @ G3 @ A3 @ X3 ) ) ) ) ) ).
% factor_def
thf(fact_446_factor__def,axiom,
( factor2800830226752492592t_unit
= ( ^ [G3: partia7496981018696276118t_unit,A3: set_list_a,B4: set_list_a] :
? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ G3 ) )
& ( B4
= ( mult_s7802724872828879953t_unit @ G3 @ A3 @ X3 ) ) ) ) ) ).
% factor_def
thf(fact_447_factor__def,axiom,
( factor4555674416838057147t_unit
= ( ^ [G3: partia1167524930811108609t_unit,A3: set_list_a,B4: set_list_a] :
? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ ( partia5178357399839081912t_unit @ G3 ) )
& ( B4
= ( mult_s5375104596032733786t_unit @ G3 @ A3 @ X3 ) ) ) ) ) ).
% factor_def
thf(fact_448_factor__def,axiom,
( factor6954119973539764400t_unit
= ( ^ [G3: partia2956882679547061052t_unit,A3: list_list_a,B4: list_list_a] :
? [X3: list_list_a] :
( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G3 ) )
& ( B4
= ( mult_l4853965630390486993t_unit @ G3 @ A3 @ X3 ) ) ) ) ) ).
% factor_def
thf(fact_449_principal__domain_Oexists__gen,axiom,
! [R: partia7496981018696276118t_unit,I3: set_set_list_a] :
( ( ring_p2468016639901664456t_unit @ R )
=> ( ( ideal_5294671857925479125t_unit @ I3 @ R )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
& ( I3
= ( cgenid9032708300698165283t_unit @ R @ X2 ) ) ) ) ) ).
% principal_domain.exists_gen
thf(fact_450_principal__domain_Oexists__gen,axiom,
! [R: partia2956882679547061052t_unit,I3: set_list_list_a] :
( ( ring_p715737262848045090t_unit @ R )
=> ( ( ideal_7391923968229085103t_unit @ I3 @ R )
=> ? [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
& ( I3
= ( cgenid24865672677839267t_unit @ R @ X2 ) ) ) ) ) ).
% principal_domain.exists_gen
thf(fact_451_principal__domain_Oexists__gen,axiom,
! [R: partia2175431115845679010xt_a_b,I3: set_a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( ideal_a_b @ I3 @ R )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
& ( I3
= ( cgenid547466209912283029xt_a_b @ R @ X2 ) ) ) ) ) ).
% principal_domain.exists_gen
thf(fact_452_principal__domain_Oexists__gen,axiom,
! [R: partia2670972154091845814t_unit,I3: set_list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( ideal_8896367198367571637t_unit @ I3 @ R )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
& ( I3
= ( cgenid9131348535277946915t_unit @ R @ X2 ) ) ) ) ) ).
% principal_domain.exists_gen
thf(fact_453_irreducible__imp__maximalideal,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P ) @ r ) ) ) ).
% irreducible_imp_maximalideal
thf(fact_454_mem__upI,axiom,
! [F: nat > a,R: partia2175431115845679010xt_a_b] :
( ! [N2: nat] : ( member_a @ ( F @ N2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ? [N3: nat] : ( bound_a @ ( zero_a_b @ R ) @ N3 @ F )
=> ( member_nat_a @ F @ ( up_a_b @ R ) ) ) ) ).
% mem_upI
thf(fact_455_mem__upI,axiom,
! [F: nat > list_a,R: partia2670972154091845814t_unit] :
( ! [N2: nat] : ( member_list_a @ ( F @ N2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ? [N3: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ N3 @ F )
=> ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_456_mem__upI,axiom,
! [F: nat > set_list_a,R: partia7496981018696276118t_unit] :
( ! [N2: nat] : ( member_set_list_a @ ( F @ N2 ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ? [N3: nat] : ( bound_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ N3 @ F )
=> ( member491565700723299188list_a @ F @ ( up_set529185716248919906t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_457_mem__upI,axiom,
! [F: nat > list_list_a,R: partia2956882679547061052t_unit] :
( ! [N2: nat] : ( member_list_list_a @ ( F @ N2 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ? [N3: nat] : ( bound_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ N3 @ F )
=> ( member8650753269014980122list_a @ F @ ( up_lis8963924889346801084t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_458_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_459_finprod__Un__disjoint,axiom,
! [A: set_nat,B2: set_nat,G: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ A
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_nat_a @ G
@ ( pi_nat_a @ B2
@ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro1280035270526425175_b_nat @ r @ G @ ( sup_sup_set_nat @ A @ B2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro1280035270526425175_b_nat @ r @ G @ A ) @ ( finpro1280035270526425175_b_nat @ r @ G @ B2 ) ) ) ) ) ) ) ) ).
% finprod_Un_disjoint
thf(fact_460_finprod__Un__disjoint,axiom,
! [A: set_a,B2: set_a,G: a > a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( ( inf_inf_set_a @ A @ B2 )
= bot_bot_set_a )
=> ( ( member_a_a @ G
@ ( pi_a_a @ A
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_a_a @ G
@ ( pi_a_a @ B2
@ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ G @ ( sup_sup_set_a @ A @ B2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ G @ A ) @ ( finpro205304725090349623_a_b_a @ r @ G @ B2 ) ) ) ) ) ) ) ) ).
% finprod_Un_disjoint
thf(fact_461_finprod__Un__disjoint,axiom,
! [A: set_list_a,B2: set_list_a,G: list_a > a] :
( ( finite_finite_list_a @ A )
=> ( ( finite_finite_list_a @ B2 )
=> ( ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ A
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a_a @ G
@ ( pi_list_a_a @ B2
@ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro6052973074229812797list_a @ r @ G @ ( sup_sup_set_list_a @ A @ B2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro6052973074229812797list_a @ r @ G @ A ) @ ( finpro6052973074229812797list_a @ r @ G @ B2 ) ) ) ) ) ) ) ) ).
% finprod_Un_disjoint
thf(fact_462_finprod__Un__disjoint,axiom,
! [A: set_set_list_a,B2: set_set_list_a,G: set_list_a > a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( finite5282473924520328461list_a @ B2 )
=> ( ( ( inf_in4657809108759609906list_a @ A @ B2 )
= bot_bo3186585308812441520list_a )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ A
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_set_list_a_a @ G
@ ( pi_set_list_a_a @ B2
@ ^ [Uu: set_list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro3826550488720007709list_a @ r @ G @ ( sup_su4537662296134749976list_a @ A @ B2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro3826550488720007709list_a @ r @ G @ A ) @ ( finpro3826550488720007709list_a @ r @ G @ B2 ) ) ) ) ) ) ) ) ).
% finprod_Un_disjoint
thf(fact_463_finprod__Un__disjoint,axiom,
! [A: set_set_a,B2: set_set_a,G: set_a > a] :
( ( finite_finite_set_a @ A )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( ( inf_inf_set_set_a @ A @ B2 )
= bot_bot_set_set_a )
=> ( ( member_set_a_a @ G
@ ( pi_set_a_a @ A
@ ^ [Uu: set_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_set_a_a @ G
@ ( pi_set_a_a @ B2
@ ^ [Uu: set_a] : ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( finpro934595834566309783_set_a @ r @ G @ ( sup_sup_set_set_a @ A @ B2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( finpro934595834566309783_set_a @ r @ G @ A ) @ ( finpro934595834566309783_set_a @ r @ G @ B2 ) ) ) ) ) ) ) ) ).
% finprod_Un_disjoint
thf(fact_464_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_465_x_Oring_Oideal__vimage,axiom,
! [I3: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ( ideal_8896367198367571637t_unit
@ ( collect_list_a
@ ^ [R4: list_a] :
( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( member_a @ ( eval_a_b @ r @ R4 @ x ) @ I3 ) ) )
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.ring.ideal_vimage
thf(fact_466_monoid__cancelI,axiom,
( ! [A6: a,B5: a,C3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C3 @ A6 )
= ( mult_a_ring_ext_a_b @ r @ C3 @ B5 ) )
=> ( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A6 = B5 ) ) ) ) )
=> ( ! [A6: a,B5: a,C3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A6 @ C3 )
= ( mult_a_ring_ext_a_b @ r @ B5 @ C3 ) )
=> ( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A6 = B5 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_467_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_468_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_469_x_Obound__upD,axiom,
! [F: nat > list_a] :
( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [N2: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N2 @ F ) ) ).
% x.bound_upD
thf(fact_470_x_Osemiring__axioms,axiom,
semiri2871908745932252451t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.semiring_axioms
thf(fact_471_x_Oabelian__monoid__axioms,axiom,
abelia226231641709521465t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.abelian_monoid_axioms
thf(fact_472_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_473_x_Oup__add__closed,axiom,
! [P: nat > list_a,Q: nat > list_a] :
( ( member_nat_list_a @ P @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_nat_list_a @ Q @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_nat_list_a
@ ^ [I: nat] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( P @ I ) @ ( Q @ I ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.up_add_closed
thf(fact_474_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_475_x_Ocgenideal__self,axiom,
! [I4: list_a] :
( ( member_list_a @ I4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ I4 @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 ) ) ) ).
% x.cgenideal_self
thf(fact_476_x_Oto__contain__is__to__divide,axiom,
! [A2: list_a,B3: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 ) )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B3 ) ) ) ) ).
% x.to_contain_is_to_divide
thf(fact_477_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_478_x_Oisgcd__divides__l,axiom,
! [A2: list_a,B3: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B3 )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( isgcd_1118609098697428327t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ A2 @ B3 ) ) ) ) ).
% x.isgcd_divides_l
thf(fact_479_x_Oisgcd__divides__r,axiom,
! [B3: list_a,A2: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 @ A2 )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( isgcd_1118609098697428327t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 @ A2 @ B3 ) ) ) ) ).
% x.isgcd_divides_r
thf(fact_480_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_481_x_Osubset__Idl__subset,axiom,
! [I3: set_list_a,H2: set_list_a] :
( ( ord_le8861187494160871172list_a @ I3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ H2 @ I3 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H2 ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I3 ) ) ) ) ).
% x.subset_Idl_subset
thf(fact_482_x_Oadd_Or__cancel,axiom,
! [A2: list_a,C: list_a,B3: 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 ) ) @ B3 @ C ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A2 = B3 ) ) ) ) ) ).
% x.add.r_cancel
thf(fact_483_x_Oadd_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) ) ) ) ) ) ).
% x.add.m_lcomm
thf(fact_484_x_Oadd_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).
% x.add.m_comm
thf(fact_485_x_Oadd_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.add.m_assoc
thf(fact_486_x_Oadd_Ol__cancel,axiom,
! [C: list_a,A2: list_a,B3: 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 @ B3 ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A2 = B3 ) ) ) ) ) ).
% x.add.l_cancel
thf(fact_487_x_Oinv__unique,axiom,
! [Y: list_a,X: list_a,Y2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% x.inv_unique
thf(fact_488_x_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X2 )
= X2 ) )
=> ( U
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.one_unique
thf(fact_489_x_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) ) ) ) ) ) ).
% x.m_lcomm
thf(fact_490_x_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).
% x.m_comm
thf(fact_491_x_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.m_assoc
thf(fact_492_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_493_x_Odivides__trans,axiom,
! [A2: list_a,B3: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B3 )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 @ 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_494_x_Oadd_Oinv__comm,axiom,
! [X: list_a,Y: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.inv_comm
thf(fact_495_x_Oadd_Ol__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.l_inv_ex
thf(fact_496_x_Oadd_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.one_unique
thf(fact_497_x_Oadd_Or__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.r_inv_ex
thf(fact_498_x_Ominus__unique,axiom,
! [Y: list_a,X: list_a,Y2: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% x.minus_unique
thf(fact_499_x_Ol__distr,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.l_distr
thf(fact_500_x_Or__distr,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ Y ) ) ) ) ) ) ).
% x.r_distr
thf(fact_501_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_502_x_Odivides__mult,axiom,
! [A2: list_a,C: list_a,B3: 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 @ B3 )
=> ( 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 @ B3 ) ) ) ) ) ).
% x.divides_mult
thf(fact_503_x_Odivides__prod__l,axiom,
! [A2: list_a,B3: list_a,C: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B3 )
=> ( 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 @ B3 ) ) ) ) ) ) ).
% x.divides_prod_l
thf(fact_504_x_Odivides__prod__r,axiom,
! [A2: list_a,B3: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B3 )
=> ( ( 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 ) ) @ B3 @ C ) ) ) ) ) ).
% x.divides_prod_r
thf(fact_505_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 ) ) ) )
=> ( ! [A6: list_a] :
( ( member_list_a @ A6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A6
!= ( 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 ) ) @ A6 @ 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_506_x_Ocgenideal__minimal,axiom,
! [J2: set_list_a,A2: list_a] :
( ( ideal_8896367198367571637t_unit @ J2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ A2 @ J2 )
=> ( ord_le8861187494160871172list_a @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 ) @ J2 ) ) ) ).
% x.cgenideal_minimal
thf(fact_507_x_Oi__intersect,axiom,
! [I3: set_list_a,J2: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ideal_8896367198367571637t_unit @ J2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ideal_8896367198367571637t_unit @ ( inf_inf_set_list_a @ I3 @ J2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.i_intersect
thf(fact_508_x_Ogenideal__minimal,axiom,
! [I3: set_list_a,S: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ S @ I3 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S ) @ I3 ) ) ) ).
% x.genideal_minimal
thf(fact_509_x_Oup__smult__closed,axiom,
! [A2: list_a,P: nat > list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_nat_list_a @ P @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_nat_list_a
@ ^ [I: nat] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ ( P @ I ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.up_smult_closed
thf(fact_510_eval__in__carrier__2,axiom,
! [X: list_a,Y: a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier_2
thf(fact_511_x_Ocgenideal__ideal,axiom,
! [A2: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ideal_8896367198367571637t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.cgenideal_ideal
thf(fact_512_x_Ooneideal,axiom,
ideal_8896367198367571637t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.oneideal
thf(fact_513_x_OIdl__subset__ideal,axiom,
! [I3: set_list_a,H2: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ H2 @ ( 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 ) ) @ H2 ) @ I3 )
= ( ord_le8861187494160871172list_a @ H2 @ I3 ) ) ) ) ).
% x.Idl_subset_ideal
thf(fact_514_x_Ogenideal__ideal,axiom,
! [S: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ideal_8896367198367571637t_unit @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.genideal_ideal
thf(fact_515_x_Onoetherian__ringI,axiom,
( ! [I5: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I5 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ? [A4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( finite_finite_list_a @ A4 )
& ( I5
= ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A4 ) ) ) )
=> ( ring_n5188127996776581661t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.noetherian_ringI
thf(fact_516_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_517_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_518_x_Oring_Ozero__closed,axiom,
member_a @ ( eval_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x ) @ ( partia707051561876973205xt_a_b @ r ) ).
% x.ring.zero_closed
thf(fact_519_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_520_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_521_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_522_x_Oadd_Oright__cancel,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% x.add.right_cancel
thf(fact_523_x_Oadd_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.m_closed
thf(fact_524_x_Ol__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% x.l_one
thf(fact_525_x_Or__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X ) ) ).
% x.r_one
thf(fact_526_x_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.m_closed
thf(fact_527_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_528_x_Oadd_Ol__cancel__one,axiom,
! [X: list_a,A2: list_a] :
( ( member_list_a @ X @ ( 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 ) ) @ X @ A2 )
= X )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one
thf(fact_529_x_Oadd_Ol__cancel__one_H,axiom,
! [X: list_a,A2: list_a] :
( ( member_list_a @ X @ ( 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
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A2 ) )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one'
thf(fact_530_x_Oadd_Or__cancel__one,axiom,
! [X: list_a,A2: list_a] :
( ( member_list_a @ X @ ( 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 @ X )
= X )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one
thf(fact_531_x_Oadd_Or__cancel__one_H,axiom,
! [X: list_a,A2: list_a] :
( ( member_list_a @ X @ ( 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
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ X ) )
= ( A2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one'
thf(fact_532_x_Ol__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% x.l_zero
thf(fact_533_x_Or__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X ) ) ).
% x.r_zero
thf(fact_534_x_Ol__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.l_null
thf(fact_535_x_Or__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.r_null
thf(fact_536_x_Odivides__mult__lI,axiom,
! [A2: list_a,B3: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B3 )
=> ( ( 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 @ B3 ) ) ) ) ) ).
% x.divides_mult_lI
thf(fact_537_x_Odivides__mult__rI,axiom,
! [A2: list_a,B3: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B3 )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ 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 ) ) @ B3 @ C ) ) ) ) ) ) ).
% x.divides_mult_rI
thf(fact_538_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_539_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_540_finprod__empty,axiom,
! [F: set_list_a > a] :
( ( finpro3826550488720007709list_a @ r @ F @ bot_bo3186585308812441520list_a )
= ( one_a_ring_ext_a_b @ r ) ) ).
% finprod_empty
thf(fact_541_finprod__empty,axiom,
! [F: set_a > a] :
( ( finpro934595834566309783_set_a @ r @ F @ bot_bot_set_set_a )
= ( one_a_ring_ext_a_b @ r ) ) ).
% finprod_empty
thf(fact_542_x_Oring_Ohom__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_a @ ( eval_a_b @ r @ X @ x ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.ring.hom_closed
thf(fact_543_x_Oring_Ohom__zero,axiom,
( ( eval_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x )
= ( zero_a_b @ r ) ) ).
% x.ring.hom_zero
thf(fact_544_x_Oring_Ohom__one,axiom,
( ( eval_a_b @ r @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x )
= ( one_a_ring_ext_a_b @ r ) ) ).
% x.ring.hom_one
thf(fact_545_x_Oring_Ohom__add,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ x )
= ( add_a_b @ r @ ( eval_a_b @ r @ X @ x ) @ ( eval_a_b @ r @ Y @ x ) ) ) ) ) ).
% x.ring.hom_add
thf(fact_546_x_Oring_Ohom__mult,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ x )
= ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ X @ x ) @ ( eval_a_b @ r @ Y @ x ) ) ) ) ) ).
% x.ring.hom_mult
thf(fact_547_monoid__cancel_Ois__monoid__cancel,axiom,
! [G2: partia2175431115845679010xt_a_b] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( monoid5798828371819920185xt_a_b @ G2 ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_548_monoid__cancel_Ois__monoid__cancel,axiom,
! [G2: partia2670972154091845814t_unit] :
( ( monoid4303264861975686087t_unit @ G2 )
=> ( monoid4303264861975686087t_unit @ G2 ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_549_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_550_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_551_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_552_finite_OemptyI,axiom,
finite5282473924520328461list_a @ bot_bo3186585308812441520list_a ).
% finite.emptyI
thf(fact_553_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_554_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_555_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_556_infinite__imp__nonempty,axiom,
! [S: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ( S != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_557_infinite__imp__nonempty,axiom,
! [S: set_set_list_a] :
( ~ ( finite5282473924520328461list_a @ S )
=> ( S != bot_bo3186585308812441520list_a ) ) ).
% infinite_imp_nonempty
thf(fact_558_infinite__imp__nonempty,axiom,
! [S: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ( S != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_559_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_560_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_561_finite__has__maximal,axiom,
! [A: set_set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( A != bot_bo3186585308812441520list_a )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_562_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_563_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_564_finite__has__minimal,axiom,
! [A: set_set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( A != bot_bo3186585308812441520list_a )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_565_monoid__cancel_Or__cancel,axiom,
! [G2: partia2175431115845679010xt_a_b,A2: a,C: a,B3: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( ( mult_a_ring_ext_a_b @ G2 @ A2 @ C )
= ( mult_a_ring_ext_a_b @ G2 @ B3 @ C ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( A2 = B3 ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_566_monoid__cancel_Or__cancel,axiom,
! [G2: partia2670972154091845814t_unit,A2: list_a,C: list_a,B3: list_a] :
( ( monoid4303264861975686087t_unit @ G2 )
=> ( ( ( mult_l7073676228092353617t_unit @ G2 @ A2 @ C )
= ( mult_l7073676228092353617t_unit @ G2 @ B3 @ C ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( A2 = B3 ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_567_monoid__cancel_Or__cancel,axiom,
! [G2: partia7496981018696276118t_unit,A2: set_list_a,C: set_list_a,B3: set_list_a] :
( ( monoid5701485383557686215t_unit @ G2 )
=> ( ( ( mult_s7802724872828879953t_unit @ G2 @ A2 @ C )
= ( mult_s7802724872828879953t_unit @ G2 @ B3 @ C ) )
=> ( ( member_set_list_a @ A2 @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( member_set_list_a @ B3 @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ G2 ) )
=> ( A2 = B3 ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_568_monoid__cancel_Or__cancel,axiom,
! [G2: partia1167524930811108609t_unit,A2: set_list_a,C: set_list_a,B3: set_list_a] :
( ( monoid6364184704960567652t_unit @ G2 )
=> ( ( ( mult_s5375104596032733786t_unit @ G2 @ A2 @ C )
= ( mult_s5375104596032733786t_unit @ G2 @ B3 @ C ) )
=> ( ( member_set_list_a @ A2 @ ( partia5178357399839081912t_unit @ G2 ) )
=> ( ( member_set_list_a @ B3 @ ( partia5178357399839081912t_unit @ G2 ) )
=> ( ( member_set_list_a @ C @ ( partia5178357399839081912t_unit @ G2 ) )
=> ( A2 = B3 ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_569_monoid__cancel_Or__cancel,axiom,
! [G2: partia2956882679547061052t_unit,A2: list_list_a,C: list_list_a,B3: list_list_a] :
( ( monoid576229335242748231t_unit @ G2 )
=> ( ( ( mult_l4853965630390486993t_unit @ G2 @ A2 @ C )
= ( mult_l4853965630390486993t_unit @ G2 @ B3 @ C ) )
=> ( ( member_list_list_a @ A2 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ B3 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( A2 = B3 ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_570_monoid__cancel_Ol__cancel,axiom,
! [G2: partia2175431115845679010xt_a_b,C: a,A2: a,B3: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( ( mult_a_ring_ext_a_b @ G2 @ C @ A2 )
= ( mult_a_ring_ext_a_b @ G2 @ C @ B3 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( A2 = B3 ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_571_monoid__cancel_Ol__cancel,axiom,
! [G2: partia2670972154091845814t_unit,C: list_a,A2: list_a,B3: list_a] :
( ( monoid4303264861975686087t_unit @ G2 )
=> ( ( ( mult_l7073676228092353617t_unit @ G2 @ C @ A2 )
= ( mult_l7073676228092353617t_unit @ G2 @ C @ B3 ) )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( A2 = B3 ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_572_monoid__cancel_Ol__cancel,axiom,
! [G2: partia7496981018696276118t_unit,C: set_list_a,A2: set_list_a,B3: set_list_a] :
( ( monoid5701485383557686215t_unit @ G2 )
=> ( ( ( mult_s7802724872828879953t_unit @ G2 @ C @ A2 )
= ( mult_s7802724872828879953t_unit @ G2 @ C @ B3 ) )
=> ( ( member_set_list_a @ A2 @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( member_set_list_a @ B3 @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ G2 ) )
=> ( A2 = B3 ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_573_monoid__cancel_Ol__cancel,axiom,
! [G2: partia1167524930811108609t_unit,C: set_list_a,A2: set_list_a,B3: set_list_a] :
( ( monoid6364184704960567652t_unit @ G2 )
=> ( ( ( mult_s5375104596032733786t_unit @ G2 @ C @ A2 )
= ( mult_s5375104596032733786t_unit @ G2 @ C @ B3 ) )
=> ( ( member_set_list_a @ A2 @ ( partia5178357399839081912t_unit @ G2 ) )
=> ( ( member_set_list_a @ B3 @ ( partia5178357399839081912t_unit @ G2 ) )
=> ( ( member_set_list_a @ C @ ( partia5178357399839081912t_unit @ G2 ) )
=> ( A2 = B3 ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_574_monoid__cancel_Ol__cancel,axiom,
! [G2: partia2956882679547061052t_unit,C: list_list_a,A2: list_list_a,B3: list_list_a] :
( ( monoid576229335242748231t_unit @ G2 )
=> ( ( ( mult_l4853965630390486993t_unit @ G2 @ C @ A2 )
= ( mult_l4853965630390486993t_unit @ G2 @ C @ B3 ) )
=> ( ( member_list_list_a @ A2 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ B3 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( A2 = B3 ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_575_ring__prime__def,axiom,
( ring_ring_prime_a_b
= ( ^ [R5: partia2175431115845679010xt_a_b,A3: a] :
( ( A3
!= ( zero_a_b @ R5 ) )
& ( prime_a_ring_ext_a_b @ R5 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_576_ring__prime__def,axiom,
( ring_r6430282645014804837t_unit
= ( ^ [R5: partia2670972154091845814t_unit,A3: list_a] :
( ( A3
!= ( zero_l4142658623432671053t_unit @ R5 ) )
& ( prime_2011924034616061926t_unit @ R5 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_577_monoid__cancel_Odivides__mult__l,axiom,
! [G2: partia2175431115845679010xt_a_b,A2: a,B3: a,C: a] :
( ( monoid5798828371819920185xt_a_b @ G2 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( factor8216151070175719842xt_a_b @ G2 @ ( mult_a_ring_ext_a_b @ G2 @ C @ A2 ) @ ( mult_a_ring_ext_a_b @ G2 @ C @ B3 ) )
= ( factor8216151070175719842xt_a_b @ G2 @ A2 @ B3 ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_578_monoid__cancel_Odivides__mult__l,axiom,
! [G2: partia2670972154091845814t_unit,A2: list_a,B3: list_a,C: list_a] :
( ( monoid4303264861975686087t_unit @ G2 )
=> ( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( factor1757716651909850160t_unit @ G2 @ ( mult_l7073676228092353617t_unit @ G2 @ C @ A2 ) @ ( mult_l7073676228092353617t_unit @ G2 @ C @ B3 ) )
= ( factor1757716651909850160t_unit @ G2 @ A2 @ B3 ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_579_monoid__cancel_Odivides__mult__l,axiom,
! [G2: partia7496981018696276118t_unit,A2: set_list_a,B3: set_list_a,C: set_list_a] :
( ( monoid5701485383557686215t_unit @ G2 )
=> ( ( member_set_list_a @ A2 @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( member_set_list_a @ B3 @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( factor2800830226752492592t_unit @ G2 @ ( mult_s7802724872828879953t_unit @ G2 @ C @ A2 ) @ ( mult_s7802724872828879953t_unit @ G2 @ C @ B3 ) )
= ( factor2800830226752492592t_unit @ G2 @ A2 @ B3 ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_580_monoid__cancel_Odivides__mult__l,axiom,
! [G2: partia1167524930811108609t_unit,A2: set_list_a,B3: set_list_a,C: set_list_a] :
( ( monoid6364184704960567652t_unit @ G2 )
=> ( ( member_set_list_a @ A2 @ ( partia5178357399839081912t_unit @ G2 ) )
=> ( ( member_set_list_a @ B3 @ ( partia5178357399839081912t_unit @ G2 ) )
=> ( ( member_set_list_a @ C @ ( partia5178357399839081912t_unit @ G2 ) )
=> ( ( factor4555674416838057147t_unit @ G2 @ ( mult_s5375104596032733786t_unit @ G2 @ C @ A2 ) @ ( mult_s5375104596032733786t_unit @ G2 @ C @ B3 ) )
= ( factor4555674416838057147t_unit @ G2 @ A2 @ B3 ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_581_monoid__cancel_Odivides__mult__l,axiom,
! [G2: partia2956882679547061052t_unit,A2: list_list_a,B3: list_list_a,C: list_list_a] :
( ( monoid576229335242748231t_unit @ G2 )
=> ( ( member_list_list_a @ A2 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ B3 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( factor6954119973539764400t_unit @ G2 @ ( mult_l4853965630390486993t_unit @ G2 @ C @ A2 ) @ ( mult_l4853965630390486993t_unit @ G2 @ C @ B3 ) )
= ( factor6954119973539764400t_unit @ G2 @ A2 @ B3 ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_582_bound__below,axiom,
! [Z: a,M2: nat,F: nat > a,N: nat] :
( ( bound_a @ Z @ M2 @ F )
=> ( ( ( F @ N )
!= Z )
=> ( ord_less_eq_nat @ N @ M2 ) ) ) ).
% bound_below
thf(fact_583_bound__below,axiom,
! [Z: list_a,M2: nat,F: nat > list_a,N: nat] :
( ( bound_list_a @ Z @ M2 @ F )
=> ( ( ( F @ N )
!= Z )
=> ( ord_less_eq_nat @ N @ M2 ) ) ) ).
% bound_below
thf(fact_584_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia7496981018696276118t_unit,P: set_list_a] :
( ( ring_p2468016639901664456t_unit @ R )
=> ( ( member_set_list_a @ P @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r5115406448772830318t_unit @ R @ P )
=> ( maxima3875439991530298004t_unit @ ( cgenid9032708300698165283t_unit @ R @ P ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_585_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( ring_p715737262848045090t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ P )
=> ( maxima7552488817642790894t_unit @ ( cgenid24865672677839267t_unit @ R @ P ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_586_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ P )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ R @ P ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_587_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ P )
=> ( maxima6585700282301356660t_unit @ ( cgenid9131348535277946915t_unit @ R @ P ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_588_mem__upD,axiom,
! [F: nat > a,R: partia2175431115845679010xt_a_b,N: nat] :
( ( member_nat_a @ F @ ( up_a_b @ R ) )
=> ( member_a @ ( F @ N ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% mem_upD
thf(fact_589_mem__upD,axiom,
! [F: nat > list_a,R: partia2670972154091845814t_unit,N: nat] :
( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) )
=> ( member_list_a @ ( F @ N ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% mem_upD
thf(fact_590_mem__upD,axiom,
! [F: nat > set_list_a,R: partia7496981018696276118t_unit,N: nat] :
( ( member491565700723299188list_a @ F @ ( up_set529185716248919906t_unit @ R ) )
=> ( member_set_list_a @ ( F @ N ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% mem_upD
thf(fact_591_mem__upD,axiom,
! [F: nat > list_list_a,R: partia2956882679547061052t_unit,N: nat] :
( ( member8650753269014980122list_a @ F @ ( up_lis8963924889346801084t_unit @ R ) )
=> ( member_list_list_a @ ( F @ N ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% mem_upD
thf(fact_592_x_Ocgenideal__is__principalideal,axiom,
! [I4: list_a] :
( ( member_list_a @ I4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.cgenideal_is_principalideal
thf(fact_593_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_594_x_Oring__hom__cring__axioms,axiom,
( ring_h1547129875642963619it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ).
% x.ring_hom_cring_axioms
thf(fact_595_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_596_x_Oring_Ois__abelian__group__hom,axiom,
( abelia8217020544048703197it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ).
% x.ring.is_abelian_group_hom
thf(fact_597_finite__number__of__roots,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( finite_finite_a @ ( collect_a @ ( polyno4133073214067823460ot_a_b @ r @ P ) ) ) ) ).
% finite_number_of_roots
thf(fact_598_Diff__disjoint,axiom,
! [A: set_set_list_a,B2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ A @ ( minus_4782336368215558443list_a @ B2 @ A ) )
= bot_bo3186585308812441520list_a ) ).
% Diff_disjoint
thf(fact_599_Diff__disjoint,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( inf_inf_set_set_a @ A @ ( minus_5736297505244876581_set_a @ B2 @ A ) )
= bot_bot_set_set_a ) ).
% Diff_disjoint
thf(fact_600_Diff__disjoint,axiom,
! [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B2 @ A ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_601_Diff__disjoint,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( minus_646659088055828811list_a @ B2 @ A ) )
= bot_bot_set_list_a ) ).
% Diff_disjoint
thf(fact_602_Diff__eq__empty__iff,axiom,
! [A: set_set_list_a,B2: set_set_list_a] :
( ( ( minus_4782336368215558443list_a @ A @ B2 )
= bot_bo3186585308812441520list_a )
= ( ord_le8877086941679407844list_a @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_603_Diff__eq__empty__iff,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A @ B2 )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_604_Diff__eq__empty__iff,axiom,
! [A: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_605_Diff__eq__empty__iff,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ( minus_646659088055828811list_a @ A @ B2 )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_606_subset__antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_607_subset__antisym,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_608_subsetI,axiom,
! [A: set_list_a_a,B2: set_list_a_a] :
( ! [X2: list_a > a] :
( ( member_list_a_a @ X2 @ A )
=> ( member_list_a_a @ X2 @ B2 ) )
=> ( ord_le6942402695062981877st_a_a @ A @ B2 ) ) ).
% subsetI
thf(fact_609_subsetI,axiom,
! [A: set_set_list_a_a,B2: set_set_list_a_a] :
( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A )
=> ( member_set_list_a_a @ X2 @ B2 ) )
=> ( ord_le4799719167512954133st_a_a @ A @ B2 ) ) ).
% subsetI
thf(fact_610_subsetI,axiom,
! [A: set_nat_list_a,B2: set_nat_list_a] :
( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( member_nat_list_a @ X2 @ B2 ) )
=> ( ord_le2145805922479659755list_a @ A @ B2 ) ) ).
% subsetI
thf(fact_611_subsetI,axiom,
! [A: set_nat_a,B2: set_nat_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( member_nat_a @ X2 @ B2 ) )
=> ( ord_le871467723717165285_nat_a @ A @ B2 ) ) ).
% subsetI
thf(fact_612_subsetI,axiom,
! [A: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% subsetI
thf(fact_613_subsetI,axiom,
! [A: set_list_a,B2: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_list_a @ X2 @ B2 ) )
=> ( ord_le8861187494160871172list_a @ A @ B2 ) ) ).
% subsetI
thf(fact_614_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X3: nat] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_615_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X3: a] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_616_empty__Collect__eq,axiom,
! [P2: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P2 ) )
= ( ! [X3: list_a] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_617_empty__Collect__eq,axiom,
! [P2: set_list_a > $o] :
( ( bot_bo3186585308812441520list_a
= ( collect_set_list_a @ P2 ) )
= ( ! [X3: set_list_a] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_618_empty__Collect__eq,axiom,
! [P2: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P2 ) )
= ( ! [X3: set_a] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_619_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_620_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_621_Collect__empty__eq,axiom,
! [P2: list_a > $o] :
( ( ( collect_list_a @ P2 )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_622_Collect__empty__eq,axiom,
! [P2: set_list_a > $o] :
( ( ( collect_set_list_a @ P2 )
= bot_bo3186585308812441520list_a )
= ( ! [X3: set_list_a] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_623_Collect__empty__eq,axiom,
! [P2: set_a > $o] :
( ( ( collect_set_a @ P2 )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_624_all__not__in__conv,axiom,
! [A: set_list_a_a] :
( ( ! [X3: list_a > a] :
~ ( member_list_a_a @ X3 @ A ) )
= ( A = bot_bot_set_list_a_a ) ) ).
% all_not_in_conv
thf(fact_625_all__not__in__conv,axiom,
! [A: set_set_list_a_a] :
( ( ! [X3: set_list_a > a] :
~ ( member_set_list_a_a @ X3 @ A ) )
= ( A = bot_bo8301825967528238409st_a_a ) ) ).
% all_not_in_conv
thf(fact_626_all__not__in__conv,axiom,
! [A: set_nat_list_a] :
( ( ! [X3: nat > list_a] :
~ ( member_nat_list_a @ X3 @ A ) )
= ( A = bot_bo3806784159821827511list_a ) ) ).
% all_not_in_conv
thf(fact_627_all__not__in__conv,axiom,
! [A: set_nat_a] :
( ( ! [X3: nat > a] :
~ ( member_nat_a @ X3 @ A ) )
= ( A = bot_bot_set_nat_a ) ) ).
% all_not_in_conv
thf(fact_628_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_629_all__not__in__conv,axiom,
! [A: set_list_a] :
( ( ! [X3: list_a] :
~ ( member_list_a @ X3 @ A ) )
= ( A = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_630_all__not__in__conv,axiom,
! [A: set_set_list_a] :
( ( ! [X3: set_list_a] :
~ ( member_set_list_a @ X3 @ A ) )
= ( A = bot_bo3186585308812441520list_a ) ) ).
% all_not_in_conv
thf(fact_631_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_632_empty__iff,axiom,
! [C: list_a > a] :
~ ( member_list_a_a @ C @ bot_bot_set_list_a_a ) ).
% empty_iff
thf(fact_633_empty__iff,axiom,
! [C: set_list_a > a] :
~ ( member_set_list_a_a @ C @ bot_bo8301825967528238409st_a_a ) ).
% empty_iff
thf(fact_634_empty__iff,axiom,
! [C: nat > list_a] :
~ ( member_nat_list_a @ C @ bot_bo3806784159821827511list_a ) ).
% empty_iff
thf(fact_635_empty__iff,axiom,
! [C: nat > a] :
~ ( member_nat_a @ C @ bot_bot_set_nat_a ) ).
% empty_iff
thf(fact_636_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_637_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_638_empty__iff,axiom,
! [C: set_list_a] :
~ ( member_set_list_a @ C @ bot_bo3186585308812441520list_a ) ).
% empty_iff
thf(fact_639_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_640_Int__iff,axiom,
! [C: list_a > a,A: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C @ ( inf_inf_set_list_a_a @ A @ B2 ) )
= ( ( member_list_a_a @ C @ A )
& ( member_list_a_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_641_Int__iff,axiom,
! [C: set_list_a > a,A: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C @ ( inf_in6568206481208318535st_a_a @ A @ B2 ) )
= ( ( member_set_list_a_a @ C @ A )
& ( member_set_list_a_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_642_Int__iff,axiom,
! [C: nat > list_a,A: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ ( inf_in6652419485960844601list_a @ A @ B2 ) )
= ( ( member_nat_list_a @ C @ A )
& ( member_nat_list_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_643_Int__iff,axiom,
! [C: nat > a,A: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( inf_inf_set_nat_a @ A @ B2 ) )
= ( ( member_nat_a @ C @ A )
& ( member_nat_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_644_Int__iff,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B2 ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_645_Int__iff,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B2 ) )
= ( ( member_list_a @ C @ A )
& ( member_list_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_646_IntI,axiom,
! [C: list_a > a,A: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C @ A )
=> ( ( member_list_a_a @ C @ B2 )
=> ( member_list_a_a @ C @ ( inf_inf_set_list_a_a @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_647_IntI,axiom,
! [C: set_list_a > a,A: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C @ A )
=> ( ( member_set_list_a_a @ C @ B2 )
=> ( member_set_list_a_a @ C @ ( inf_in6568206481208318535st_a_a @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_648_IntI,axiom,
! [C: nat > list_a,A: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ A )
=> ( ( member_nat_list_a @ C @ B2 )
=> ( member_nat_list_a @ C @ ( inf_in6652419485960844601list_a @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_649_IntI,axiom,
! [C: nat > a,A: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ A )
=> ( ( member_nat_a @ C @ B2 )
=> ( member_nat_a @ C @ ( inf_inf_set_nat_a @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_650_IntI,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_651_IntI,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A )
=> ( ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_652_Diff__idemp,axiom,
! [A: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B2 ) @ B2 )
= ( minus_minus_set_a @ A @ B2 ) ) ).
% Diff_idemp
thf(fact_653_Diff__idemp,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A @ B2 ) @ B2 )
= ( minus_646659088055828811list_a @ A @ B2 ) ) ).
% Diff_idemp
thf(fact_654_Diff__iff,axiom,
! [C: list_a > a,A: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C @ ( minus_921748639838131438st_a_a @ A @ B2 ) )
= ( ( member_list_a_a @ C @ A )
& ~ ( member_list_a_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_655_Diff__iff,axiom,
! [C: set_list_a > a,A: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C @ ( minus_5613498140476352782st_a_a @ A @ B2 ) )
= ( ( member_set_list_a_a @ C @ A )
& ~ ( member_set_list_a_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_656_Diff__iff,axiom,
! [C: nat > list_a,A: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ ( minus_4169782841487898290list_a @ A @ B2 ) )
= ( ( member_nat_list_a @ C @ A )
& ~ ( member_nat_list_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_657_Diff__iff,axiom,
! [C: nat > a,A: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A @ B2 ) )
= ( ( member_nat_a @ C @ A )
& ~ ( member_nat_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_658_Diff__iff,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_659_Diff__iff,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) )
= ( ( member_list_a @ C @ A )
& ~ ( member_list_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_660_DiffI,axiom,
! [C: list_a > a,A: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C @ A )
=> ( ~ ( member_list_a_a @ C @ B2 )
=> ( member_list_a_a @ C @ ( minus_921748639838131438st_a_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_661_DiffI,axiom,
! [C: set_list_a > a,A: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C @ A )
=> ( ~ ( member_set_list_a_a @ C @ B2 )
=> ( member_set_list_a_a @ C @ ( minus_5613498140476352782st_a_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_662_DiffI,axiom,
! [C: nat > list_a,A: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ A )
=> ( ~ ( member_nat_list_a @ C @ B2 )
=> ( member_nat_list_a @ C @ ( minus_4169782841487898290list_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_663_DiffI,axiom,
! [C: nat > a,A: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ A )
=> ( ~ ( member_nat_a @ C @ B2 )
=> ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_664_DiffI,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_665_DiffI,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A )
=> ( ~ ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_666_Un__iff,axiom,
! [C: list_a > a,A: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C @ ( sup_sup_set_list_a_a @ A @ B2 ) )
= ( ( member_list_a_a @ C @ A )
| ( member_list_a_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_667_Un__iff,axiom,
! [C: set_list_a > a,A: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C @ ( sup_su8833226608330050529st_a_a @ A @ B2 ) )
= ( ( member_set_list_a_a @ C @ A )
| ( member_set_list_a_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_668_Un__iff,axiom,
! [C: nat > list_a,A: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ ( sup_su5649930751583389983list_a @ A @ B2 ) )
= ( ( member_nat_list_a @ C @ A )
| ( member_nat_list_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_669_Un__iff,axiom,
! [C: nat > a,A: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( sup_sup_set_nat_a @ A @ B2 ) )
= ( ( member_nat_a @ C @ A )
| ( member_nat_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_670_Un__iff,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B2 ) )
= ( ( member_list_a @ C @ A )
| ( member_list_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_671_Un__iff,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B2 ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_672_UnCI,axiom,
! [C: list_a > a,B2: set_list_a_a,A: set_list_a_a] :
( ( ~ ( member_list_a_a @ C @ B2 )
=> ( member_list_a_a @ C @ A ) )
=> ( member_list_a_a @ C @ ( sup_sup_set_list_a_a @ A @ B2 ) ) ) ).
% UnCI
thf(fact_673_UnCI,axiom,
! [C: set_list_a > a,B2: set_set_list_a_a,A: set_set_list_a_a] :
( ( ~ ( member_set_list_a_a @ C @ B2 )
=> ( member_set_list_a_a @ C @ A ) )
=> ( member_set_list_a_a @ C @ ( sup_su8833226608330050529st_a_a @ A @ B2 ) ) ) ).
% UnCI
thf(fact_674_UnCI,axiom,
! [C: nat > list_a,B2: set_nat_list_a,A: set_nat_list_a] :
( ( ~ ( member_nat_list_a @ C @ B2 )
=> ( member_nat_list_a @ C @ A ) )
=> ( member_nat_list_a @ C @ ( sup_su5649930751583389983list_a @ A @ B2 ) ) ) ).
% UnCI
thf(fact_675_UnCI,axiom,
! [C: nat > a,B2: set_nat_a,A: set_nat_a] :
( ( ~ ( member_nat_a @ C @ B2 )
=> ( member_nat_a @ C @ A ) )
=> ( member_nat_a @ C @ ( sup_sup_set_nat_a @ A @ B2 ) ) ) ).
% UnCI
thf(fact_676_UnCI,axiom,
! [C: list_a,B2: set_list_a,A: set_list_a] :
( ( ~ ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ A ) )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B2 ) ) ) ).
% UnCI
thf(fact_677_UnCI,axiom,
! [C: a,B2: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% UnCI
thf(fact_678_x_Omonoid__cancelI,axiom,
( ! [A6: list_a,B5: list_a,C3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 @ A6 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 @ B5 ) )
=> ( ( member_list_a @ A6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A6 = B5 ) ) ) ) )
=> ( ! [A6: list_a,B5: list_a,C3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A6 @ C3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B5 @ C3 ) )
=> ( ( member_list_a @ A6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A6 = B5 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.monoid_cancelI
thf(fact_679_is__root__poly__mult__imp__is__root,axiom,
! [P: list_a,Q: list_a,X: 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 ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_680_empty__subsetI,axiom,
! [A: set_set_list_a] : ( ord_le8877086941679407844list_a @ bot_bo3186585308812441520list_a @ A ) ).
% empty_subsetI
thf(fact_681_empty__subsetI,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% empty_subsetI
thf(fact_682_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_683_empty__subsetI,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A ) ).
% empty_subsetI
thf(fact_684_subset__empty,axiom,
! [A: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A @ bot_bo3186585308812441520list_a )
= ( A = bot_bo3186585308812441520list_a ) ) ).
% subset_empty
thf(fact_685_subset__empty,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_686_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_687_subset__empty,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
= ( A = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_688_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B2 ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_689_Int__subset__iff,axiom,
! [C2: set_list_a,A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ ( inf_inf_set_list_a @ A @ B2 ) )
= ( ( ord_le8861187494160871172list_a @ C2 @ A )
& ( ord_le8861187494160871172list_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_690_Diff__cancel,axiom,
! [A: set_set_list_a] :
( ( minus_4782336368215558443list_a @ A @ A )
= bot_bo3186585308812441520list_a ) ).
% Diff_cancel
thf(fact_691_Diff__cancel,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ A )
= bot_bot_set_set_a ) ).
% Diff_cancel
thf(fact_692_Diff__cancel,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_693_Diff__cancel,axiom,
! [A: set_list_a] :
( ( minus_646659088055828811list_a @ A @ A )
= bot_bot_set_list_a ) ).
% Diff_cancel
thf(fact_694_empty__Diff,axiom,
! [A: set_set_list_a] :
( ( minus_4782336368215558443list_a @ bot_bo3186585308812441520list_a @ A )
= bot_bo3186585308812441520list_a ) ).
% empty_Diff
thf(fact_695_empty__Diff,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ bot_bot_set_set_a @ A )
= bot_bot_set_set_a ) ).
% empty_Diff
thf(fact_696_empty__Diff,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_697_empty__Diff,axiom,
! [A: set_list_a] :
( ( minus_646659088055828811list_a @ bot_bot_set_list_a @ A )
= bot_bot_set_list_a ) ).
% empty_Diff
thf(fact_698_Diff__empty,axiom,
! [A: set_set_list_a] :
( ( minus_4782336368215558443list_a @ A @ bot_bo3186585308812441520list_a )
= A ) ).
% Diff_empty
thf(fact_699_Diff__empty,axiom,
! [A: set_set_a] :
( ( minus_5736297505244876581_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% Diff_empty
thf(fact_700_Diff__empty,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ) ).
% Diff_empty
thf(fact_701_Diff__empty,axiom,
! [A: set_list_a] :
( ( minus_646659088055828811list_a @ A @ bot_bot_set_list_a )
= A ) ).
% Diff_empty
thf(fact_702_Un__subset__iff,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ C2 )
= ( ( ord_less_eq_set_a @ A @ C2 )
& ( ord_less_eq_set_a @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_703_Un__subset__iff,axiom,
! [A: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B2 ) @ C2 )
= ( ( ord_le8861187494160871172list_a @ A @ C2 )
& ( ord_le8861187494160871172list_a @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_704_Un__empty,axiom,
! [A: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_705_Un__empty,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ( sup_sup_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
= ( ( A = bot_bot_set_list_a )
& ( B2 = bot_bot_set_list_a ) ) ) ).
% Un_empty
thf(fact_706_Un__empty,axiom,
! [A: set_set_list_a,B2: set_set_list_a] :
( ( ( sup_su4537662296134749976list_a @ A @ B2 )
= bot_bo3186585308812441520list_a )
= ( ( A = bot_bo3186585308812441520list_a )
& ( B2 = bot_bo3186585308812441520list_a ) ) ) ).
% Un_empty
thf(fact_707_Un__empty,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ( sup_sup_set_set_a @ A @ B2 )
= bot_bot_set_set_a )
= ( ( A = bot_bot_set_set_a )
& ( B2 = bot_bot_set_set_a ) ) ) ).
% Un_empty
thf(fact_708_Un__Int__eq_I1_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_709_Un__Int__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_710_Un__Int__eq_I2_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( inf_inf_set_list_a @ ( sup_sup_set_list_a @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_711_Un__Int__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_712_Un__Int__eq_I3_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( inf_inf_set_list_a @ S @ ( sup_sup_set_list_a @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_713_Un__Int__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_714_Un__Int__eq_I4_J,axiom,
! [T: set_list_a,S: set_list_a] :
( ( inf_inf_set_list_a @ T @ ( sup_sup_set_list_a @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_715_Un__Int__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( inf_inf_set_a @ T @ ( sup_sup_set_a @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_716_Int__Un__eq_I1_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_717_Int__Un__eq_I1_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_718_Int__Un__eq_I2_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( sup_sup_set_list_a @ ( inf_inf_set_list_a @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_719_Int__Un__eq_I2_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_720_Int__Un__eq_I3_J,axiom,
! [S: set_list_a,T: set_list_a] :
( ( sup_sup_set_list_a @ S @ ( inf_inf_set_list_a @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_721_Int__Un__eq_I3_J,axiom,
! [S: set_a,T: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_722_Int__Un__eq_I4_J,axiom,
! [T: set_list_a,S: set_list_a] :
( ( sup_sup_set_list_a @ T @ ( inf_inf_set_list_a @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_723_Int__Un__eq_I4_J,axiom,
! [T: set_a,S: set_a] :
( ( sup_sup_set_a @ T @ ( inf_inf_set_a @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_724_Un__Diff__cancel2,axiom,
! [B2: set_a,A: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B2 @ A ) @ A )
= ( sup_sup_set_a @ B2 @ A ) ) ).
% Un_Diff_cancel2
thf(fact_725_Un__Diff__cancel2,axiom,
! [B2: set_list_a,A: set_list_a] :
( ( sup_sup_set_list_a @ ( minus_646659088055828811list_a @ B2 @ A ) @ A )
= ( sup_sup_set_list_a @ B2 @ A ) ) ).
% Un_Diff_cancel2
thf(fact_726_Un__Diff__cancel,axiom,
! [A: set_a,B2: set_a] :
( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B2 @ A ) )
= ( sup_sup_set_a @ A @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_727_Un__Diff__cancel,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( sup_sup_set_list_a @ A @ ( minus_646659088055828811list_a @ B2 @ A ) )
= ( sup_sup_set_list_a @ A @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_728_ring__hom__cring_Ohom__zero,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H: a > a] :
( ( ring_h661254511236296859_b_a_b @ R @ S @ H )
=> ( ( H @ ( zero_a_b @ R ) )
= ( zero_a_b @ S ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_729_ring__hom__cring_Ohom__zero,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H: a > list_a] :
( ( ring_h8279546866833948963t_unit @ R @ S @ H )
=> ( ( H @ ( zero_a_b @ R ) )
= ( zero_l4142658623432671053t_unit @ S ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_730_ring__hom__cring_Ohom__zero,axiom,
! [R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H: list_a > list_a] :
( ( ring_h8282015026914974507t_unit @ R @ S @ H )
=> ( ( H @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ S ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_731_ring__hom__cring_Ohom__zero,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H: list_a > a] :
( ( ring_h1547129875642963619it_a_b @ R @ S @ H )
=> ( ( H @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_a_b @ S ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_732_Collect__mono__iff,axiom,
! [P2: set_list_a > $o,Q2: set_list_a > $o] :
( ( ord_le8877086941679407844list_a @ ( collect_set_list_a @ P2 ) @ ( collect_set_list_a @ Q2 ) )
= ( ! [X3: set_list_a] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_733_Collect__mono__iff,axiom,
! [P2: set_a > $o,Q2: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P2 ) @ ( collect_set_a @ Q2 ) )
= ( ! [X3: set_a] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_734_Collect__mono__iff,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) )
= ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_735_Collect__mono__iff,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) )
= ( ! [X3: a] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_736_Collect__mono__iff,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q2 ) )
= ( ! [X3: list_a] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_737_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A7: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A7 @ B )
& ( ord_less_eq_set_a @ B @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_738_set__eq__subset,axiom,
( ( ^ [Y5: set_list_a,Z3: set_list_a] : ( Y5 = Z3 ) )
= ( ^ [A7: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A7 @ B )
& ( ord_le8861187494160871172list_a @ B @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_739_subset__trans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_740_subset__trans,axiom,
! [A: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C2 )
=> ( ord_le8861187494160871172list_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_741_Collect__mono,axiom,
! [P2: set_list_a > $o,Q2: set_list_a > $o] :
( ! [X2: set_list_a] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) )
=> ( ord_le8877086941679407844list_a @ ( collect_set_list_a @ P2 ) @ ( collect_set_list_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_742_Collect__mono,axiom,
! [P2: set_a > $o,Q2: set_a > $o] :
( ! [X2: set_a] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P2 ) @ ( collect_set_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_743_Collect__mono,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ! [X2: nat] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_mono
thf(fact_744_Collect__mono,axiom,
! [P2: a > $o,Q2: a > $o] :
( ! [X2: a] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_745_Collect__mono,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ! [X2: list_a] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_746_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_747_subset__refl,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).
% subset_refl
thf(fact_748_subset__iff,axiom,
( ord_le6942402695062981877st_a_a
= ( ^ [A7: set_list_a_a,B: set_list_a_a] :
! [T2: list_a > a] :
( ( member_list_a_a @ T2 @ A7 )
=> ( member_list_a_a @ T2 @ B ) ) ) ) ).
% subset_iff
thf(fact_749_subset__iff,axiom,
( ord_le4799719167512954133st_a_a
= ( ^ [A7: set_set_list_a_a,B: set_set_list_a_a] :
! [T2: set_list_a > a] :
( ( member_set_list_a_a @ T2 @ A7 )
=> ( member_set_list_a_a @ T2 @ B ) ) ) ) ).
% subset_iff
thf(fact_750_subset__iff,axiom,
( ord_le2145805922479659755list_a
= ( ^ [A7: set_nat_list_a,B: set_nat_list_a] :
! [T2: nat > list_a] :
( ( member_nat_list_a @ T2 @ A7 )
=> ( member_nat_list_a @ T2 @ B ) ) ) ) ).
% subset_iff
thf(fact_751_subset__iff,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A7: set_nat_a,B: set_nat_a] :
! [T2: nat > a] :
( ( member_nat_a @ T2 @ A7 )
=> ( member_nat_a @ T2 @ B ) ) ) ) ).
% subset_iff
thf(fact_752_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A7 )
=> ( member_a @ T2 @ B ) ) ) ) ).
% subset_iff
thf(fact_753_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
! [T2: list_a] :
( ( member_list_a @ T2 @ A7 )
=> ( member_list_a @ T2 @ B ) ) ) ) ).
% subset_iff
thf(fact_754_equalityD2,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ).
% equalityD2
thf(fact_755_equalityD2,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( A = B2 )
=> ( ord_le8861187494160871172list_a @ B2 @ A ) ) ).
% equalityD2
thf(fact_756_equalityD1,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% equalityD1
thf(fact_757_equalityD1,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( A = B2 )
=> ( ord_le8861187494160871172list_a @ A @ B2 ) ) ).
% equalityD1
thf(fact_758_subset__eq,axiom,
( ord_le6942402695062981877st_a_a
= ( ^ [A7: set_list_a_a,B: set_list_a_a] :
! [X3: list_a > a] :
( ( member_list_a_a @ X3 @ A7 )
=> ( member_list_a_a @ X3 @ B ) ) ) ) ).
% subset_eq
thf(fact_759_subset__eq,axiom,
( ord_le4799719167512954133st_a_a
= ( ^ [A7: set_set_list_a_a,B: set_set_list_a_a] :
! [X3: set_list_a > a] :
( ( member_set_list_a_a @ X3 @ A7 )
=> ( member_set_list_a_a @ X3 @ B ) ) ) ) ).
% subset_eq
thf(fact_760_subset__eq,axiom,
( ord_le2145805922479659755list_a
= ( ^ [A7: set_nat_list_a,B: set_nat_list_a] :
! [X3: nat > list_a] :
( ( member_nat_list_a @ X3 @ A7 )
=> ( member_nat_list_a @ X3 @ B ) ) ) ) ).
% subset_eq
thf(fact_761_subset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A7: set_nat_a,B: set_nat_a] :
! [X3: nat > a] :
( ( member_nat_a @ X3 @ A7 )
=> ( member_nat_a @ X3 @ B ) ) ) ) ).
% subset_eq
thf(fact_762_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A7 )
=> ( member_a @ X3 @ B ) ) ) ) ).
% subset_eq
thf(fact_763_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ A7 )
=> ( member_list_a @ X3 @ B ) ) ) ) ).
% subset_eq
thf(fact_764_equalityE,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ~ ( ( ord_less_eq_set_a @ A @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A ) ) ) ).
% equalityE
thf(fact_765_equalityE,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( A = B2 )
=> ~ ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ~ ( ord_le8861187494160871172list_a @ B2 @ A ) ) ) ).
% equalityE
thf(fact_766_subsetD,axiom,
! [A: set_list_a_a,B2: set_list_a_a,C: list_a > a] :
( ( ord_le6942402695062981877st_a_a @ A @ B2 )
=> ( ( member_list_a_a @ C @ A )
=> ( member_list_a_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_767_subsetD,axiom,
! [A: set_set_list_a_a,B2: set_set_list_a_a,C: set_list_a > a] :
( ( ord_le4799719167512954133st_a_a @ A @ B2 )
=> ( ( member_set_list_a_a @ C @ A )
=> ( member_set_list_a_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_768_subsetD,axiom,
! [A: set_nat_list_a,B2: set_nat_list_a,C: nat > list_a] :
( ( ord_le2145805922479659755list_a @ A @ B2 )
=> ( ( member_nat_list_a @ C @ A )
=> ( member_nat_list_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_769_subsetD,axiom,
! [A: set_nat_a,B2: set_nat_a,C: nat > a] :
( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( ( member_nat_a @ C @ A )
=> ( member_nat_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_770_subsetD,axiom,
! [A: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_771_subsetD,axiom,
! [A: set_list_a,B2: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_772_in__mono,axiom,
! [A: set_list_a_a,B2: set_list_a_a,X: list_a > a] :
( ( ord_le6942402695062981877st_a_a @ A @ B2 )
=> ( ( member_list_a_a @ X @ A )
=> ( member_list_a_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_773_in__mono,axiom,
! [A: set_set_list_a_a,B2: set_set_list_a_a,X: set_list_a > a] :
( ( ord_le4799719167512954133st_a_a @ A @ B2 )
=> ( ( member_set_list_a_a @ X @ A )
=> ( member_set_list_a_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_774_in__mono,axiom,
! [A: set_nat_list_a,B2: set_nat_list_a,X: nat > list_a] :
( ( ord_le2145805922479659755list_a @ A @ B2 )
=> ( ( member_nat_list_a @ X @ A )
=> ( member_nat_list_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_775_in__mono,axiom,
! [A: set_nat_a,B2: set_nat_a,X: nat > a] :
( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( ( member_nat_a @ X @ A )
=> ( member_nat_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_776_in__mono,axiom,
! [A: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_777_in__mono,axiom,
! [A: set_list_a,B2: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( member_list_a @ X @ A )
=> ( member_list_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_778_ex__in__conv,axiom,
! [A: set_list_a_a] :
( ( ? [X3: list_a > a] : ( member_list_a_a @ X3 @ A ) )
= ( A != bot_bot_set_list_a_a ) ) ).
% ex_in_conv
thf(fact_779_ex__in__conv,axiom,
! [A: set_set_list_a_a] :
( ( ? [X3: set_list_a > a] : ( member_set_list_a_a @ X3 @ A ) )
= ( A != bot_bo8301825967528238409st_a_a ) ) ).
% ex_in_conv
thf(fact_780_ex__in__conv,axiom,
! [A: set_nat_list_a] :
( ( ? [X3: nat > list_a] : ( member_nat_list_a @ X3 @ A ) )
= ( A != bot_bo3806784159821827511list_a ) ) ).
% ex_in_conv
thf(fact_781_ex__in__conv,axiom,
! [A: set_nat_a] :
( ( ? [X3: nat > a] : ( member_nat_a @ X3 @ A ) )
= ( A != bot_bot_set_nat_a ) ) ).
% ex_in_conv
thf(fact_782_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_783_ex__in__conv,axiom,
! [A: set_list_a] :
( ( ? [X3: list_a] : ( member_list_a @ X3 @ A ) )
= ( A != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_784_ex__in__conv,axiom,
! [A: set_set_list_a] :
( ( ? [X3: set_list_a] : ( member_set_list_a @ X3 @ A ) )
= ( A != bot_bo3186585308812441520list_a ) ) ).
% ex_in_conv
thf(fact_785_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_786_equals0I,axiom,
! [A: set_list_a_a] :
( ! [Y4: list_a > a] :
~ ( member_list_a_a @ Y4 @ A )
=> ( A = bot_bot_set_list_a_a ) ) ).
% equals0I
thf(fact_787_equals0I,axiom,
! [A: set_set_list_a_a] :
( ! [Y4: set_list_a > a] :
~ ( member_set_list_a_a @ Y4 @ A )
=> ( A = bot_bo8301825967528238409st_a_a ) ) ).
% equals0I
thf(fact_788_equals0I,axiom,
! [A: set_nat_list_a] :
( ! [Y4: nat > list_a] :
~ ( member_nat_list_a @ Y4 @ A )
=> ( A = bot_bo3806784159821827511list_a ) ) ).
% equals0I
thf(fact_789_equals0I,axiom,
! [A: set_nat_a] :
( ! [Y4: nat > a] :
~ ( member_nat_a @ Y4 @ A )
=> ( A = bot_bot_set_nat_a ) ) ).
% equals0I
thf(fact_790_equals0I,axiom,
! [A: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_791_equals0I,axiom,
! [A: set_list_a] :
( ! [Y4: list_a] :
~ ( member_list_a @ Y4 @ A )
=> ( A = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_792_equals0I,axiom,
! [A: set_set_list_a] :
( ! [Y4: set_list_a] :
~ ( member_set_list_a @ Y4 @ A )
=> ( A = bot_bo3186585308812441520list_a ) ) ).
% equals0I
thf(fact_793_equals0I,axiom,
! [A: set_set_a] :
( ! [Y4: set_a] :
~ ( member_set_a @ Y4 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_794_equals0D,axiom,
! [A: set_list_a_a,A2: list_a > a] :
( ( A = bot_bot_set_list_a_a )
=> ~ ( member_list_a_a @ A2 @ A ) ) ).
% equals0D
thf(fact_795_equals0D,axiom,
! [A: set_set_list_a_a,A2: set_list_a > a] :
( ( A = bot_bo8301825967528238409st_a_a )
=> ~ ( member_set_list_a_a @ A2 @ A ) ) ).
% equals0D
thf(fact_796_equals0D,axiom,
! [A: set_nat_list_a,A2: nat > list_a] :
( ( A = bot_bo3806784159821827511list_a )
=> ~ ( member_nat_list_a @ A2 @ A ) ) ).
% equals0D
thf(fact_797_equals0D,axiom,
! [A: set_nat_a,A2: nat > a] :
( ( A = bot_bot_set_nat_a )
=> ~ ( member_nat_a @ A2 @ A ) ) ).
% equals0D
thf(fact_798_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_799_equals0D,axiom,
! [A: set_list_a,A2: list_a] :
( ( A = bot_bot_set_list_a )
=> ~ ( member_list_a @ A2 @ A ) ) ).
% equals0D
thf(fact_800_equals0D,axiom,
! [A: set_set_list_a,A2: set_list_a] :
( ( A = bot_bo3186585308812441520list_a )
=> ~ ( member_set_list_a @ A2 @ A ) ) ).
% equals0D
thf(fact_801_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_802_emptyE,axiom,
! [A2: list_a > a] :
~ ( member_list_a_a @ A2 @ bot_bot_set_list_a_a ) ).
% emptyE
thf(fact_803_emptyE,axiom,
! [A2: set_list_a > a] :
~ ( member_set_list_a_a @ A2 @ bot_bo8301825967528238409st_a_a ) ).
% emptyE
thf(fact_804_emptyE,axiom,
! [A2: nat > list_a] :
~ ( member_nat_list_a @ A2 @ bot_bo3806784159821827511list_a ) ).
% emptyE
thf(fact_805_emptyE,axiom,
! [A2: nat > a] :
~ ( member_nat_a @ A2 @ bot_bot_set_nat_a ) ).
% emptyE
thf(fact_806_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_807_emptyE,axiom,
! [A2: list_a] :
~ ( member_list_a @ A2 @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_808_emptyE,axiom,
! [A2: set_list_a] :
~ ( member_set_list_a @ A2 @ bot_bo3186585308812441520list_a ) ).
% emptyE
thf(fact_809_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_810_Int__left__commute,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_811_Int__left__commute,axiom,
! [A: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B2 @ C2 ) )
= ( inf_inf_set_list_a @ B2 @ ( inf_inf_set_list_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_812_Int__left__absorb,axiom,
! [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B2 ) )
= ( inf_inf_set_a @ A @ B2 ) ) ).
% Int_left_absorb
thf(fact_813_Int__left__absorb,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ A @ B2 ) )
= ( inf_inf_set_list_a @ A @ B2 ) ) ).
% Int_left_absorb
thf(fact_814_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A7: set_a,B: set_a] : ( inf_inf_set_a @ B @ A7 ) ) ) ).
% Int_commute
thf(fact_815_Int__commute,axiom,
( inf_inf_set_list_a
= ( ^ [A7: set_list_a,B: set_list_a] : ( inf_inf_set_list_a @ B @ A7 ) ) ) ).
% Int_commute
thf(fact_816_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_817_Int__absorb,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_818_Int__assoc,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_819_Int__assoc,axiom,
! [A: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_820_IntD2,axiom,
! [C: list_a > a,A: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C @ ( inf_inf_set_list_a_a @ A @ B2 ) )
=> ( member_list_a_a @ C @ B2 ) ) ).
% IntD2
thf(fact_821_IntD2,axiom,
! [C: set_list_a > a,A: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C @ ( inf_in6568206481208318535st_a_a @ A @ B2 ) )
=> ( member_set_list_a_a @ C @ B2 ) ) ).
% IntD2
thf(fact_822_IntD2,axiom,
! [C: nat > list_a,A: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ ( inf_in6652419485960844601list_a @ A @ B2 ) )
=> ( member_nat_list_a @ C @ B2 ) ) ).
% IntD2
thf(fact_823_IntD2,axiom,
! [C: nat > a,A: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( inf_inf_set_nat_a @ A @ B2 ) )
=> ( member_nat_a @ C @ B2 ) ) ).
% IntD2
thf(fact_824_IntD2,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B2 ) )
=> ( member_a @ C @ B2 ) ) ).
% IntD2
thf(fact_825_IntD2,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B2 ) )
=> ( member_list_a @ C @ B2 ) ) ).
% IntD2
thf(fact_826_IntD1,axiom,
! [C: list_a > a,A: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C @ ( inf_inf_set_list_a_a @ A @ B2 ) )
=> ( member_list_a_a @ C @ A ) ) ).
% IntD1
thf(fact_827_IntD1,axiom,
! [C: set_list_a > a,A: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C @ ( inf_in6568206481208318535st_a_a @ A @ B2 ) )
=> ( member_set_list_a_a @ C @ A ) ) ).
% IntD1
thf(fact_828_IntD1,axiom,
! [C: nat > list_a,A: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ ( inf_in6652419485960844601list_a @ A @ B2 ) )
=> ( member_nat_list_a @ C @ A ) ) ).
% IntD1
thf(fact_829_IntD1,axiom,
! [C: nat > a,A: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( inf_inf_set_nat_a @ A @ B2 ) )
=> ( member_nat_a @ C @ A ) ) ).
% IntD1
thf(fact_830_IntD1,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B2 ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_831_IntD1,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B2 ) )
=> ( member_list_a @ C @ A ) ) ).
% IntD1
thf(fact_832_IntE,axiom,
! [C: list_a > a,A: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C @ ( inf_inf_set_list_a_a @ A @ B2 ) )
=> ~ ( ( member_list_a_a @ C @ A )
=> ~ ( member_list_a_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_833_IntE,axiom,
! [C: set_list_a > a,A: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C @ ( inf_in6568206481208318535st_a_a @ A @ B2 ) )
=> ~ ( ( member_set_list_a_a @ C @ A )
=> ~ ( member_set_list_a_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_834_IntE,axiom,
! [C: nat > list_a,A: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ ( inf_in6652419485960844601list_a @ A @ B2 ) )
=> ~ ( ( member_nat_list_a @ C @ A )
=> ~ ( member_nat_list_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_835_IntE,axiom,
! [C: nat > a,A: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( inf_inf_set_nat_a @ A @ B2 ) )
=> ~ ( ( member_nat_a @ C @ A )
=> ~ ( member_nat_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_836_IntE,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B2 ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_837_IntE,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B2 ) )
=> ~ ( ( member_list_a @ C @ A )
=> ~ ( member_list_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_838_DiffD2,axiom,
! [C: list_a > a,A: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C @ ( minus_921748639838131438st_a_a @ A @ B2 ) )
=> ~ ( member_list_a_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_839_DiffD2,axiom,
! [C: set_list_a > a,A: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C @ ( minus_5613498140476352782st_a_a @ A @ B2 ) )
=> ~ ( member_set_list_a_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_840_DiffD2,axiom,
! [C: nat > list_a,A: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ ( minus_4169782841487898290list_a @ A @ B2 ) )
=> ~ ( member_nat_list_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_841_DiffD2,axiom,
! [C: nat > a,A: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A @ B2 ) )
=> ~ ( member_nat_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_842_DiffD2,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_843_DiffD2,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) )
=> ~ ( member_list_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_844_DiffD1,axiom,
! [C: list_a > a,A: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C @ ( minus_921748639838131438st_a_a @ A @ B2 ) )
=> ( member_list_a_a @ C @ A ) ) ).
% DiffD1
thf(fact_845_DiffD1,axiom,
! [C: set_list_a > a,A: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C @ ( minus_5613498140476352782st_a_a @ A @ B2 ) )
=> ( member_set_list_a_a @ C @ A ) ) ).
% DiffD1
thf(fact_846_DiffD1,axiom,
! [C: nat > list_a,A: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ ( minus_4169782841487898290list_a @ A @ B2 ) )
=> ( member_nat_list_a @ C @ A ) ) ).
% DiffD1
thf(fact_847_DiffD1,axiom,
! [C: nat > a,A: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A @ B2 ) )
=> ( member_nat_a @ C @ A ) ) ).
% DiffD1
thf(fact_848_DiffD1,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_849_DiffD1,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) )
=> ( member_list_a @ C @ A ) ) ).
% DiffD1
thf(fact_850_DiffE,axiom,
! [C: list_a > a,A: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C @ ( minus_921748639838131438st_a_a @ A @ B2 ) )
=> ~ ( ( member_list_a_a @ C @ A )
=> ( member_list_a_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_851_DiffE,axiom,
! [C: set_list_a > a,A: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C @ ( minus_5613498140476352782st_a_a @ A @ B2 ) )
=> ~ ( ( member_set_list_a_a @ C @ A )
=> ( member_set_list_a_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_852_DiffE,axiom,
! [C: nat > list_a,A: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ ( minus_4169782841487898290list_a @ A @ B2 ) )
=> ~ ( ( member_nat_list_a @ C @ A )
=> ( member_nat_list_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_853_DiffE,axiom,
! [C: nat > a,A: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A @ B2 ) )
=> ~ ( ( member_nat_a @ C @ A )
=> ( member_nat_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_854_DiffE,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_855_DiffE,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) )
=> ~ ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_856_Un__left__commute,axiom,
! [A: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( sup_sup_set_list_a @ A @ ( sup_sup_set_list_a @ B2 @ C2 ) )
= ( sup_sup_set_list_a @ B2 @ ( sup_sup_set_list_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_857_Un__left__commute,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B2 @ C2 ) )
= ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_858_Un__left__absorb,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( sup_sup_set_list_a @ A @ ( sup_sup_set_list_a @ A @ B2 ) )
= ( sup_sup_set_list_a @ A @ B2 ) ) ).
% Un_left_absorb
thf(fact_859_Un__left__absorb,axiom,
! [A: set_a,B2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B2 ) )
= ( sup_sup_set_a @ A @ B2 ) ) ).
% Un_left_absorb
thf(fact_860_Un__commute,axiom,
( sup_sup_set_list_a
= ( ^ [A7: set_list_a,B: set_list_a] : ( sup_sup_set_list_a @ B @ A7 ) ) ) ).
% Un_commute
thf(fact_861_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A7: set_a,B: set_a] : ( sup_sup_set_a @ B @ A7 ) ) ) ).
% Un_commute
thf(fact_862_Un__absorb,axiom,
! [A: set_list_a] :
( ( sup_sup_set_list_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_863_Un__absorb,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% Un_absorb
thf(fact_864_Un__assoc,axiom,
! [A: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( sup_sup_set_list_a @ ( sup_sup_set_list_a @ A @ B2 ) @ C2 )
= ( sup_sup_set_list_a @ A @ ( sup_sup_set_list_a @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_865_Un__assoc,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B2 ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_866_ball__Un,axiom,
! [A: set_list_a,B2: set_list_a,P2: list_a > $o] :
( ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( sup_sup_set_list_a @ A @ B2 ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ( P2 @ X3 ) )
& ! [X3: list_a] :
( ( member_list_a @ X3 @ B2 )
=> ( P2 @ X3 ) ) ) ) ).
% ball_Un
thf(fact_867_ball__Un,axiom,
! [A: set_a,B2: set_a,P2: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A @ B2 ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( P2 @ X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( P2 @ X3 ) ) ) ) ).
% ball_Un
thf(fact_868_bex__Un,axiom,
! [A: set_list_a,B2: set_list_a,P2: list_a > $o] :
( ( ? [X3: list_a] :
( ( member_list_a @ X3 @ ( sup_sup_set_list_a @ A @ B2 ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ A )
& ( P2 @ X3 ) )
| ? [X3: list_a] :
( ( member_list_a @ X3 @ B2 )
& ( P2 @ X3 ) ) ) ) ).
% bex_Un
thf(fact_869_bex__Un,axiom,
! [A: set_a,B2: set_a,P2: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A @ B2 ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A )
& ( P2 @ X3 ) )
| ? [X3: a] :
( ( member_a @ X3 @ B2 )
& ( P2 @ X3 ) ) ) ) ).
% bex_Un
thf(fact_870_UnI2,axiom,
! [C: list_a > a,B2: set_list_a_a,A: set_list_a_a] :
( ( member_list_a_a @ C @ B2 )
=> ( member_list_a_a @ C @ ( sup_sup_set_list_a_a @ A @ B2 ) ) ) ).
% UnI2
thf(fact_871_UnI2,axiom,
! [C: set_list_a > a,B2: set_set_list_a_a,A: set_set_list_a_a] :
( ( member_set_list_a_a @ C @ B2 )
=> ( member_set_list_a_a @ C @ ( sup_su8833226608330050529st_a_a @ A @ B2 ) ) ) ).
% UnI2
thf(fact_872_UnI2,axiom,
! [C: nat > list_a,B2: set_nat_list_a,A: set_nat_list_a] :
( ( member_nat_list_a @ C @ B2 )
=> ( member_nat_list_a @ C @ ( sup_su5649930751583389983list_a @ A @ B2 ) ) ) ).
% UnI2
thf(fact_873_UnI2,axiom,
! [C: nat > a,B2: set_nat_a,A: set_nat_a] :
( ( member_nat_a @ C @ B2 )
=> ( member_nat_a @ C @ ( sup_sup_set_nat_a @ A @ B2 ) ) ) ).
% UnI2
thf(fact_874_UnI2,axiom,
! [C: list_a,B2: set_list_a,A: set_list_a] :
( ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B2 ) ) ) ).
% UnI2
thf(fact_875_UnI2,axiom,
! [C: a,B2: set_a,A: set_a] :
( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% UnI2
thf(fact_876_UnI1,axiom,
! [C: list_a > a,A: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C @ A )
=> ( member_list_a_a @ C @ ( sup_sup_set_list_a_a @ A @ B2 ) ) ) ).
% UnI1
thf(fact_877_UnI1,axiom,
! [C: set_list_a > a,A: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C @ A )
=> ( member_set_list_a_a @ C @ ( sup_su8833226608330050529st_a_a @ A @ B2 ) ) ) ).
% UnI1
thf(fact_878_UnI1,axiom,
! [C: nat > list_a,A: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ A )
=> ( member_nat_list_a @ C @ ( sup_su5649930751583389983list_a @ A @ B2 ) ) ) ).
% UnI1
thf(fact_879_UnI1,axiom,
! [C: nat > a,A: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ A )
=> ( member_nat_a @ C @ ( sup_sup_set_nat_a @ A @ B2 ) ) ) ).
% UnI1
thf(fact_880_UnI1,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B2 ) ) ) ).
% UnI1
thf(fact_881_UnI1,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% UnI1
thf(fact_882_UnE,axiom,
! [C: list_a > a,A: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C @ ( sup_sup_set_list_a_a @ A @ B2 ) )
=> ( ~ ( member_list_a_a @ C @ A )
=> ( member_list_a_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_883_UnE,axiom,
! [C: set_list_a > a,A: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C @ ( sup_su8833226608330050529st_a_a @ A @ B2 ) )
=> ( ~ ( member_set_list_a_a @ C @ A )
=> ( member_set_list_a_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_884_UnE,axiom,
! [C: nat > list_a,A: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ ( sup_su5649930751583389983list_a @ A @ B2 ) )
=> ( ~ ( member_nat_list_a @ C @ A )
=> ( member_nat_list_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_885_UnE,axiom,
! [C: nat > a,A: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( sup_sup_set_nat_a @ A @ B2 ) )
=> ( ~ ( member_nat_a @ C @ A )
=> ( member_nat_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_886_UnE,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( sup_sup_set_list_a @ A @ B2 ) )
=> ( ~ ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_887_UnE,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B2 ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_888_Collect__subset,axiom,
! [A: set_list_a_a,P2: ( list_a > a ) > $o] :
( ord_le6942402695062981877st_a_a
@ ( collect_list_a_a
@ ^ [X3: list_a > a] :
( ( member_list_a_a @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_889_Collect__subset,axiom,
! [A: set_set_list_a_a,P2: ( set_list_a > a ) > $o] :
( ord_le4799719167512954133st_a_a
@ ( collect_set_list_a_a
@ ^ [X3: set_list_a > a] :
( ( member_set_list_a_a @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_890_Collect__subset,axiom,
! [A: set_nat_list_a,P2: ( nat > list_a ) > $o] :
( ord_le2145805922479659755list_a
@ ( collect_nat_list_a
@ ^ [X3: nat > list_a] :
( ( member_nat_list_a @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_891_Collect__subset,axiom,
! [A: set_nat_a,P2: ( nat > a ) > $o] :
( ord_le871467723717165285_nat_a
@ ( collect_nat_a
@ ^ [X3: nat > a] :
( ( member_nat_a @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_892_Collect__subset,axiom,
! [A: set_set_list_a,P2: set_list_a > $o] :
( ord_le8877086941679407844list_a
@ ( collect_set_list_a
@ ^ [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_893_Collect__subset,axiom,
! [A: set_set_a,P2: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_894_Collect__subset,axiom,
! [A: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_895_Collect__subset,axiom,
! [A: set_a,P2: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_896_Collect__subset,axiom,
! [A: set_list_a,P2: list_a > $o] :
( ord_le8861187494160871172list_a
@ ( collect_list_a
@ ^ [X3: list_a] :
( ( member_list_a @ X3 @ A )
& ( P2 @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_897_less__eq__set__def,axiom,
( ord_le6942402695062981877st_a_a
= ( ^ [A7: set_list_a_a,B: set_list_a_a] :
( ord_le5538412863658560464_a_a_o
@ ^ [X3: list_a > a] : ( member_list_a_a @ X3 @ A7 )
@ ^ [X3: list_a > a] : ( member_list_a_a @ X3 @ B ) ) ) ) ).
% less_eq_set_def
thf(fact_898_less__eq__set__def,axiom,
( ord_le4799719167512954133st_a_a
= ( ^ [A7: set_set_list_a_a,B: set_set_list_a_a] :
( ord_le6553425858663066544_a_a_o
@ ^ [X3: set_list_a > a] : ( member_set_list_a_a @ X3 @ A7 )
@ ^ [X3: set_list_a > a] : ( member_set_list_a_a @ X3 @ B ) ) ) ) ).
% less_eq_set_def
thf(fact_899_less__eq__set__def,axiom,
( ord_le2145805922479659755list_a
= ( ^ [A7: set_nat_list_a,B: set_nat_list_a] :
( ord_le4184171100712167858st_a_o
@ ^ [X3: nat > list_a] : ( member_nat_list_a @ X3 @ A7 )
@ ^ [X3: nat > list_a] : ( member_nat_list_a @ X3 @ B ) ) ) ) ).
% less_eq_set_def
thf(fact_900_less__eq__set__def,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A7: set_nat_a,B: set_nat_a] :
( ord_less_eq_nat_a_o
@ ^ [X3: nat > a] : ( member_nat_a @ X3 @ A7 )
@ ^ [X3: nat > a] : ( member_nat_a @ X3 @ B ) ) ) ) ).
% less_eq_set_def
thf(fact_901_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B: set_a] :
( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A7 )
@ ^ [X3: a] : ( member_a @ X3 @ B ) ) ) ) ).
% less_eq_set_def
thf(fact_902_less__eq__set__def,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
( ord_less_eq_list_a_o
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A7 )
@ ^ [X3: list_a] : ( member_list_a @ X3 @ B ) ) ) ) ).
% less_eq_set_def
thf(fact_903_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% empty_def
thf(fact_904_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X3: a] : $false ) ) ).
% empty_def
thf(fact_905_empty__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a
@ ^ [X3: list_a] : $false ) ) ).
% empty_def
thf(fact_906_empty__def,axiom,
( bot_bo3186585308812441520list_a
= ( collect_set_list_a
@ ^ [X3: set_list_a] : $false ) ) ).
% empty_def
thf(fact_907_empty__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a
@ ^ [X3: set_a] : $false ) ) ).
% empty_def
thf(fact_908_Collect__conj__eq,axiom,
! [P2: set_list_a > $o,Q2: set_list_a > $o] :
( ( collect_set_list_a
@ ^ [X3: set_list_a] :
( ( P2 @ X3 )
& ( Q2 @ X3 ) ) )
= ( inf_in4657809108759609906list_a @ ( collect_set_list_a @ P2 ) @ ( collect_set_list_a @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_909_Collect__conj__eq,axiom,
! [P2: set_a > $o,Q2: set_a > $o] :
( ( collect_set_a
@ ^ [X3: set_a] :
( ( P2 @ X3 )
& ( Q2 @ X3 ) ) )
= ( inf_inf_set_set_a @ ( collect_set_a @ P2 ) @ ( collect_set_a @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_910_Collect__conj__eq,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P2 @ X3 )
& ( Q2 @ X3 ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_911_Collect__conj__eq,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( P2 @ X3 )
& ( Q2 @ X3 ) ) )
= ( inf_inf_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_912_Collect__conj__eq,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( collect_list_a
@ ^ [X3: list_a] :
( ( P2 @ X3 )
& ( Q2 @ X3 ) ) )
= ( inf_inf_set_list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_913_Int__Collect,axiom,
! [X: list_a > a,A: set_list_a_a,P2: ( list_a > a ) > $o] :
( ( member_list_a_a @ X @ ( inf_inf_set_list_a_a @ A @ ( collect_list_a_a @ P2 ) ) )
= ( ( member_list_a_a @ X @ A )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_914_Int__Collect,axiom,
! [X: set_list_a > a,A: set_set_list_a_a,P2: ( set_list_a > a ) > $o] :
( ( member_set_list_a_a @ X @ ( inf_in6568206481208318535st_a_a @ A @ ( collect_set_list_a_a @ P2 ) ) )
= ( ( member_set_list_a_a @ X @ A )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_915_Int__Collect,axiom,
! [X: nat > list_a,A: set_nat_list_a,P2: ( nat > list_a ) > $o] :
( ( member_nat_list_a @ X @ ( inf_in6652419485960844601list_a @ A @ ( collect_nat_list_a @ P2 ) ) )
= ( ( member_nat_list_a @ X @ A )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_916_Int__Collect,axiom,
! [X: nat > a,A: set_nat_a,P2: ( nat > a ) > $o] :
( ( member_nat_a @ X @ ( inf_inf_set_nat_a @ A @ ( collect_nat_a @ P2 ) ) )
= ( ( member_nat_a @ X @ A )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_917_Int__Collect,axiom,
! [X: set_list_a,A: set_set_list_a,P2: set_list_a > $o] :
( ( member_set_list_a @ X @ ( inf_in4657809108759609906list_a @ A @ ( collect_set_list_a @ P2 ) ) )
= ( ( member_set_list_a @ X @ A )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_918_Int__Collect,axiom,
! [X: set_a,A: set_set_a,P2: set_a > $o] :
( ( member_set_a @ X @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P2 ) ) )
= ( ( member_set_a @ X @ A )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_919_Int__Collect,axiom,
! [X: nat,A: set_nat,P2: nat > $o] :
( ( member_nat @ X @ ( inf_inf_set_nat @ A @ ( collect_nat @ P2 ) ) )
= ( ( member_nat @ X @ A )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_920_Int__Collect,axiom,
! [X: a,A: set_a,P2: a > $o] :
( ( member_a @ X @ ( inf_inf_set_a @ A @ ( collect_a @ P2 ) ) )
= ( ( member_a @ X @ A )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_921_Int__Collect,axiom,
! [X: list_a,A: set_list_a,P2: list_a > $o] :
( ( member_list_a @ X @ ( inf_inf_set_list_a @ A @ ( collect_list_a @ P2 ) ) )
= ( ( member_list_a @ X @ A )
& ( P2 @ X ) ) ) ).
% Int_Collect
thf(fact_922_Int__def,axiom,
( inf_inf_set_list_a_a
= ( ^ [A7: set_list_a_a,B: set_list_a_a] :
( collect_list_a_a
@ ^ [X3: list_a > a] :
( ( member_list_a_a @ X3 @ A7 )
& ( member_list_a_a @ X3 @ B ) ) ) ) ) ).
% Int_def
thf(fact_923_Int__def,axiom,
( inf_in6568206481208318535st_a_a
= ( ^ [A7: set_set_list_a_a,B: set_set_list_a_a] :
( collect_set_list_a_a
@ ^ [X3: set_list_a > a] :
( ( member_set_list_a_a @ X3 @ A7 )
& ( member_set_list_a_a @ X3 @ B ) ) ) ) ) ).
% Int_def
thf(fact_924_Int__def,axiom,
( inf_in6652419485960844601list_a
= ( ^ [A7: set_nat_list_a,B: set_nat_list_a] :
( collect_nat_list_a
@ ^ [X3: nat > list_a] :
( ( member_nat_list_a @ X3 @ A7 )
& ( member_nat_list_a @ X3 @ B ) ) ) ) ) ).
% Int_def
thf(fact_925_Int__def,axiom,
( inf_inf_set_nat_a
= ( ^ [A7: set_nat_a,B: set_nat_a] :
( collect_nat_a
@ ^ [X3: nat > a] :
( ( member_nat_a @ X3 @ A7 )
& ( member_nat_a @ X3 @ B ) ) ) ) ) ).
% Int_def
thf(fact_926_Int__def,axiom,
( inf_in4657809108759609906list_a
= ( ^ [A7: set_set_list_a,B: set_set_list_a] :
( collect_set_list_a
@ ^ [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A7 )
& ( member_set_list_a @ X3 @ B ) ) ) ) ) ).
% Int_def
thf(fact_927_Int__def,axiom,
( inf_inf_set_set_a
= ( ^ [A7: set_set_a,B: set_set_a] :
( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ A7 )
& ( member_set_a @ X3 @ B ) ) ) ) ) ).
% Int_def
thf(fact_928_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A7: set_nat,B: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A7 )
& ( member_nat @ X3 @ B ) ) ) ) ) ).
% Int_def
thf(fact_929_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A7: set_a,B: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A7 )
& ( member_a @ X3 @ B ) ) ) ) ) ).
% Int_def
thf(fact_930_Int__def,axiom,
( inf_inf_set_list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
( collect_list_a
@ ^ [X3: list_a] :
( ( member_list_a @ X3 @ A7 )
& ( member_list_a @ X3 @ B ) ) ) ) ) ).
% Int_def
thf(fact_931_set__diff__eq,axiom,
( minus_921748639838131438st_a_a
= ( ^ [A7: set_list_a_a,B: set_list_a_a] :
( collect_list_a_a
@ ^ [X3: list_a > a] :
( ( member_list_a_a @ X3 @ A7 )
& ~ ( member_list_a_a @ X3 @ B ) ) ) ) ) ).
% set_diff_eq
thf(fact_932_set__diff__eq,axiom,
( minus_5613498140476352782st_a_a
= ( ^ [A7: set_set_list_a_a,B: set_set_list_a_a] :
( collect_set_list_a_a
@ ^ [X3: set_list_a > a] :
( ( member_set_list_a_a @ X3 @ A7 )
& ~ ( member_set_list_a_a @ X3 @ B ) ) ) ) ) ).
% set_diff_eq
thf(fact_933_set__diff__eq,axiom,
( minus_4169782841487898290list_a
= ( ^ [A7: set_nat_list_a,B: set_nat_list_a] :
( collect_nat_list_a
@ ^ [X3: nat > list_a] :
( ( member_nat_list_a @ X3 @ A7 )
& ~ ( member_nat_list_a @ X3 @ B ) ) ) ) ) ).
% set_diff_eq
thf(fact_934_set__diff__eq,axiom,
( minus_490503922182417452_nat_a
= ( ^ [A7: set_nat_a,B: set_nat_a] :
( collect_nat_a
@ ^ [X3: nat > a] :
( ( member_nat_a @ X3 @ A7 )
& ~ ( member_nat_a @ X3 @ B ) ) ) ) ) ).
% set_diff_eq
thf(fact_935_set__diff__eq,axiom,
( minus_4782336368215558443list_a
= ( ^ [A7: set_set_list_a,B: set_set_list_a] :
( collect_set_list_a
@ ^ [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A7 )
& ~ ( member_set_list_a @ X3 @ B ) ) ) ) ) ).
% set_diff_eq
thf(fact_936_set__diff__eq,axiom,
( minus_5736297505244876581_set_a
= ( ^ [A7: set_set_a,B: set_set_a] :
( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ A7 )
& ~ ( member_set_a @ X3 @ B ) ) ) ) ) ).
% set_diff_eq
thf(fact_937_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A7: set_nat,B: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A7 )
& ~ ( member_nat @ X3 @ B ) ) ) ) ) ).
% set_diff_eq
thf(fact_938_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A7: set_a,B: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A7 )
& ~ ( member_a @ X3 @ B ) ) ) ) ) ).
% set_diff_eq
thf(fact_939_set__diff__eq,axiom,
( minus_646659088055828811list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
( collect_list_a
@ ^ [X3: list_a] :
( ( member_list_a @ X3 @ A7 )
& ~ ( member_list_a @ X3 @ B ) ) ) ) ) ).
% set_diff_eq
thf(fact_940_Collect__disj__eq,axiom,
! [P2: set_list_a > $o,Q2: set_list_a > $o] :
( ( collect_set_list_a
@ ^ [X3: set_list_a] :
( ( P2 @ X3 )
| ( Q2 @ X3 ) ) )
= ( sup_su4537662296134749976list_a @ ( collect_set_list_a @ P2 ) @ ( collect_set_list_a @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_941_Collect__disj__eq,axiom,
! [P2: set_a > $o,Q2: set_a > $o] :
( ( collect_set_a
@ ^ [X3: set_a] :
( ( P2 @ X3 )
| ( Q2 @ X3 ) ) )
= ( sup_sup_set_set_a @ ( collect_set_a @ P2 ) @ ( collect_set_a @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_942_Collect__disj__eq,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( P2 @ X3 )
| ( Q2 @ X3 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_943_Collect__disj__eq,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( collect_list_a
@ ^ [X3: list_a] :
( ( P2 @ X3 )
| ( Q2 @ X3 ) ) )
= ( sup_sup_set_list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_944_Collect__disj__eq,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( P2 @ X3 )
| ( Q2 @ X3 ) ) )
= ( sup_sup_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_945_Un__def,axiom,
( sup_sup_set_list_a_a
= ( ^ [A7: set_list_a_a,B: set_list_a_a] :
( collect_list_a_a
@ ^ [X3: list_a > a] :
( ( member_list_a_a @ X3 @ A7 )
| ( member_list_a_a @ X3 @ B ) ) ) ) ) ).
% Un_def
thf(fact_946_Un__def,axiom,
( sup_su8833226608330050529st_a_a
= ( ^ [A7: set_set_list_a_a,B: set_set_list_a_a] :
( collect_set_list_a_a
@ ^ [X3: set_list_a > a] :
( ( member_set_list_a_a @ X3 @ A7 )
| ( member_set_list_a_a @ X3 @ B ) ) ) ) ) ).
% Un_def
thf(fact_947_Un__def,axiom,
( sup_su5649930751583389983list_a
= ( ^ [A7: set_nat_list_a,B: set_nat_list_a] :
( collect_nat_list_a
@ ^ [X3: nat > list_a] :
( ( member_nat_list_a @ X3 @ A7 )
| ( member_nat_list_a @ X3 @ B ) ) ) ) ) ).
% Un_def
thf(fact_948_Un__def,axiom,
( sup_sup_set_nat_a
= ( ^ [A7: set_nat_a,B: set_nat_a] :
( collect_nat_a
@ ^ [X3: nat > a] :
( ( member_nat_a @ X3 @ A7 )
| ( member_nat_a @ X3 @ B ) ) ) ) ) ).
% Un_def
thf(fact_949_Un__def,axiom,
( sup_su4537662296134749976list_a
= ( ^ [A7: set_set_list_a,B: set_set_list_a] :
( collect_set_list_a
@ ^ [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A7 )
| ( member_set_list_a @ X3 @ B ) ) ) ) ) ).
% Un_def
thf(fact_950_Un__def,axiom,
( sup_sup_set_set_a
= ( ^ [A7: set_set_a,B: set_set_a] :
( collect_set_a
@ ^ [X3: set_a] :
( ( member_set_a @ X3 @ A7 )
| ( member_set_a @ X3 @ B ) ) ) ) ) ).
% Un_def
thf(fact_951_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A7: set_nat,B: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A7 )
| ( member_nat @ X3 @ B ) ) ) ) ) ).
% Un_def
thf(fact_952_Un__def,axiom,
( sup_sup_set_list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
( collect_list_a
@ ^ [X3: list_a] :
( ( member_list_a @ X3 @ A7 )
| ( member_list_a @ X3 @ B ) ) ) ) ) ).
% Un_def
thf(fact_953_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A7: set_a,B: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A7 )
| ( member_a @ X3 @ B ) ) ) ) ) ).
% Un_def
thf(fact_954_Int__Collect__mono,axiom,
! [A: set_list_a_a,B2: set_list_a_a,P2: ( list_a > a ) > $o,Q2: ( list_a > a ) > $o] :
( ( ord_le6942402695062981877st_a_a @ A @ B2 )
=> ( ! [X2: list_a > a] :
( ( member_list_a_a @ X2 @ A )
=> ( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_le6942402695062981877st_a_a @ ( inf_inf_set_list_a_a @ A @ ( collect_list_a_a @ P2 ) ) @ ( inf_inf_set_list_a_a @ B2 @ ( collect_list_a_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_955_Int__Collect__mono,axiom,
! [A: set_set_list_a_a,B2: set_set_list_a_a,P2: ( set_list_a > a ) > $o,Q2: ( set_list_a > a ) > $o] :
( ( ord_le4799719167512954133st_a_a @ A @ B2 )
=> ( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A )
=> ( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_le4799719167512954133st_a_a @ ( inf_in6568206481208318535st_a_a @ A @ ( collect_set_list_a_a @ P2 ) ) @ ( inf_in6568206481208318535st_a_a @ B2 @ ( collect_set_list_a_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_956_Int__Collect__mono,axiom,
! [A: set_nat_list_a,B2: set_nat_list_a,P2: ( nat > list_a ) > $o,Q2: ( nat > list_a ) > $o] :
( ( ord_le2145805922479659755list_a @ A @ B2 )
=> ( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_le2145805922479659755list_a @ ( inf_in6652419485960844601list_a @ A @ ( collect_nat_list_a @ P2 ) ) @ ( inf_in6652419485960844601list_a @ B2 @ ( collect_nat_list_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_957_Int__Collect__mono,axiom,
! [A: set_nat_a,B2: set_nat_a,P2: ( nat > a ) > $o,Q2: ( nat > a ) > $o] :
( ( ord_le871467723717165285_nat_a @ A @ B2 )
=> ( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_le871467723717165285_nat_a @ ( inf_inf_set_nat_a @ A @ ( collect_nat_a @ P2 ) ) @ ( inf_inf_set_nat_a @ B2 @ ( collect_nat_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_958_Int__Collect__mono,axiom,
! [A: set_set_list_a,B2: set_set_list_a,P2: set_list_a > $o,Q2: set_list_a > $o] :
( ( ord_le8877086941679407844list_a @ A @ B2 )
=> ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
=> ( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_le8877086941679407844list_a @ ( inf_in4657809108759609906list_a @ A @ ( collect_set_list_a @ P2 ) ) @ ( inf_in4657809108759609906list_a @ B2 @ ( collect_set_list_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_959_Int__Collect__mono,axiom,
! [A: set_set_a,B2: set_set_a,P2: set_a > $o,Q2: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P2 ) ) @ ( inf_inf_set_set_a @ B2 @ ( collect_set_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_960_Int__Collect__mono,axiom,
! [A: set_nat,B2: set_nat,P2: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P2 ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_961_Int__Collect__mono,axiom,
! [A: set_a,B2: set_a,P2: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P2 ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_962_Int__Collect__mono,axiom,
! [A: set_list_a,B2: set_list_a,P2: list_a > $o,Q2: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ ( collect_list_a @ P2 ) ) @ ( inf_inf_set_list_a @ B2 @ ( collect_list_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_963_Int__greatest,axiom,
! [C2: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_964_Int__greatest,axiom,
! [C2: set_list_a,A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ A )
=> ( ( ord_le8861187494160871172list_a @ C2 @ B2 )
=> ( ord_le8861187494160871172list_a @ C2 @ ( inf_inf_set_list_a @ A @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_965_Int__absorb2,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( inf_inf_set_a @ A @ B2 )
= A ) ) ).
% Int_absorb2
thf(fact_966_Int__absorb2,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( inf_inf_set_list_a @ A @ B2 )
= A ) ) ).
% Int_absorb2
thf(fact_967_Int__absorb1,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( inf_inf_set_a @ A @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_968_Int__absorb1,axiom,
! [B2: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B2 @ A )
=> ( ( inf_inf_set_list_a @ A @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_969_Int__lower2,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_970_Int__lower2,axiom,
! [A: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_971_Int__lower1,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ A ) ).
% Int_lower1
thf(fact_972_Int__lower1,axiom,
! [A: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ A ) ).
% Int_lower1
thf(fact_973_Int__mono,axiom,
! [A: set_a,C2: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_974_Int__mono,axiom,
! [A: set_list_a,C2: set_list_a,B2: set_list_a,D: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ D )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ ( inf_inf_set_list_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_975_double__diff,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_976_double__diff,axiom,
! [A: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C2 )
=> ( ( minus_646659088055828811list_a @ B2 @ ( minus_646659088055828811list_a @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_977_Diff__subset,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B2 ) @ A ) ).
% Diff_subset
thf(fact_978_Diff__subset,axiom,
! [A: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A @ B2 ) @ A ) ).
% Diff_subset
thf(fact_979_Diff__mono,axiom,
! [A: set_a,C2: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B2 ) @ ( minus_minus_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_980_Diff__mono,axiom,
! [A: set_list_a,C2: set_list_a,D: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C2 )
=> ( ( ord_le8861187494160871172list_a @ D @ B2 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A @ B2 ) @ ( minus_646659088055828811list_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_981_Int__emptyI,axiom,
! [A: set_list_a_a,B2: set_list_a_a] :
( ! [X2: list_a > a] :
( ( member_list_a_a @ X2 @ A )
=> ~ ( member_list_a_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_list_a_a @ A @ B2 )
= bot_bot_set_list_a_a ) ) ).
% Int_emptyI
thf(fact_982_Int__emptyI,axiom,
! [A: set_set_list_a_a,B2: set_set_list_a_a] :
( ! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A )
=> ~ ( member_set_list_a_a @ X2 @ B2 ) )
=> ( ( inf_in6568206481208318535st_a_a @ A @ B2 )
= bot_bo8301825967528238409st_a_a ) ) ).
% Int_emptyI
thf(fact_983_Int__emptyI,axiom,
! [A: set_nat_list_a,B2: set_nat_list_a] :
( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A )
=> ~ ( member_nat_list_a @ X2 @ B2 ) )
=> ( ( inf_in6652419485960844601list_a @ A @ B2 )
= bot_bo3806784159821827511list_a ) ) ).
% Int_emptyI
thf(fact_984_Int__emptyI,axiom,
! [A: set_nat_a,B2: set_nat_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A )
=> ~ ( member_nat_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_nat_a @ A @ B2 )
= bot_bot_set_nat_a ) ) ).
% Int_emptyI
thf(fact_985_Int__emptyI,axiom,
! [A: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_a @ A @ B2 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_986_Int__emptyI,axiom,
! [A: set_list_a,B2: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ~ ( member_list_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a ) ) ).
% Int_emptyI
thf(fact_987_Int__emptyI,axiom,
! [A: set_set_list_a,B2: set_set_list_a] :
( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A )
=> ~ ( member_set_list_a @ X2 @ B2 ) )
=> ( ( inf_in4657809108759609906list_a @ A @ B2 )
= bot_bo3186585308812441520list_a ) ) ).
% Int_emptyI
thf(fact_988_Int__emptyI,axiom,
! [A: set_set_a,B2: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ~ ( member_set_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_set_a @ A @ B2 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_989_disjoint__iff,axiom,
! [A: set_list_a_a,B2: set_list_a_a] :
( ( ( inf_inf_set_list_a_a @ A @ B2 )
= bot_bot_set_list_a_a )
= ( ! [X3: list_a > a] :
( ( member_list_a_a @ X3 @ A )
=> ~ ( member_list_a_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_990_disjoint__iff,axiom,
! [A: set_set_list_a_a,B2: set_set_list_a_a] :
( ( ( inf_in6568206481208318535st_a_a @ A @ B2 )
= bot_bo8301825967528238409st_a_a )
= ( ! [X3: set_list_a > a] :
( ( member_set_list_a_a @ X3 @ A )
=> ~ ( member_set_list_a_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_991_disjoint__iff,axiom,
! [A: set_nat_list_a,B2: set_nat_list_a] :
( ( ( inf_in6652419485960844601list_a @ A @ B2 )
= bot_bo3806784159821827511list_a )
= ( ! [X3: nat > list_a] :
( ( member_nat_list_a @ X3 @ A )
=> ~ ( member_nat_list_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_992_disjoint__iff,axiom,
! [A: set_nat_a,B2: set_nat_a] :
( ( ( inf_inf_set_nat_a @ A @ B2 )
= bot_bot_set_nat_a )
= ( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ A )
=> ~ ( member_nat_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_993_disjoint__iff,axiom,
! [A: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_994_disjoint__iff,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ~ ( member_list_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_995_disjoint__iff,axiom,
! [A: set_set_list_a,B2: set_set_list_a] :
( ( ( inf_in4657809108759609906list_a @ A @ B2 )
= bot_bo3186585308812441520list_a )
= ( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A )
=> ~ ( member_set_list_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_996_disjoint__iff,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B2 )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ( member_set_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_997_Int__empty__left,axiom,
! [B2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B2 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_998_Int__empty__left,axiom,
! [B2: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ B2 )
= bot_bot_set_list_a ) ).
% Int_empty_left
thf(fact_999_Int__empty__left,axiom,
! [B2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ bot_bo3186585308812441520list_a @ B2 )
= bot_bo3186585308812441520list_a ) ).
% Int_empty_left
thf(fact_1000_Int__empty__left,axiom,
! [B2: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B2 )
= bot_bot_set_set_a ) ).
% Int_empty_left
thf(fact_1001_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_1002_Int__empty__right,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% Int_empty_right
thf(fact_1003_Int__empty__right,axiom,
! [A: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ A @ bot_bo3186585308812441520list_a )
= bot_bo3186585308812441520list_a ) ).
% Int_empty_right
thf(fact_1004_Int__empty__right,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% Int_empty_right
thf(fact_1005_disjoint__iff__not__equal,axiom,
! [A: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1006_disjoint__iff__not__equal,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B2 )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ! [Y3: list_a] :
( ( member_list_a @ Y3 @ B2 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1007_disjoint__iff__not__equal,axiom,
! [A: set_set_list_a,B2: set_set_list_a] :
( ( ( inf_in4657809108759609906list_a @ A @ B2 )
= bot_bo3186585308812441520list_a )
= ( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A )
=> ! [Y3: set_list_a] :
( ( member_set_list_a @ Y3 @ B2 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1008_disjoint__iff__not__equal,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B2 )
= bot_bot_set_set_a )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ! [Y3: set_a] :
( ( member_set_a @ Y3 @ B2 )
=> ( X3 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1009_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A7: set_a,B: set_a] :
( ( sup_sup_set_a @ A7 @ B )
= B ) ) ) ).
% subset_Un_eq
thf(fact_1010_subset__Un__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A7: set_list_a,B: set_list_a] :
( ( sup_sup_set_list_a @ A7 @ B )
= B ) ) ) ).
% subset_Un_eq
thf(fact_1011_subset__UnE,axiom,
! [C2: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B2 ) )
=> ~ ! [A8: set_a] :
( ( ord_less_eq_set_a @ A8 @ A )
=> ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ B2 )
=> ( C2
!= ( sup_sup_set_a @ A8 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1012_subset__UnE,axiom,
! [C2: set_list_a,A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ ( sup_sup_set_list_a @ A @ B2 ) )
=> ~ ! [A8: set_list_a] :
( ( ord_le8861187494160871172list_a @ A8 @ A )
=> ! [B6: set_list_a] :
( ( ord_le8861187494160871172list_a @ B6 @ B2 )
=> ( C2
!= ( sup_sup_set_list_a @ A8 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_1013_Un__absorb2,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( sup_sup_set_a @ A @ B2 )
= A ) ) ).
% Un_absorb2
thf(fact_1014_Un__absorb2,axiom,
! [B2: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B2 @ A )
=> ( ( sup_sup_set_list_a @ A @ B2 )
= A ) ) ).
% Un_absorb2
thf(fact_1015_Un__absorb1,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( sup_sup_set_a @ A @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_1016_Un__absorb1,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( sup_sup_set_list_a @ A @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_1017_Un__upper2,axiom,
! [B2: set_a,A: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A @ B2 ) ) ).
% Un_upper2
thf(fact_1018_Un__upper2,axiom,
! [B2: set_list_a,A: set_list_a] : ( ord_le8861187494160871172list_a @ B2 @ ( sup_sup_set_list_a @ A @ B2 ) ) ).
% Un_upper2
thf(fact_1019_Un__upper1,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B2 ) ) ).
% Un_upper1
thf(fact_1020_Un__upper1,axiom,
! [A: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ A @ ( sup_sup_set_list_a @ A @ B2 ) ) ).
% Un_upper1
thf(fact_1021_Un__least,axiom,
! [A: set_a,C2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_1022_Un__least,axiom,
! [A: set_list_a,C2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C2 )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_1023_Un__mono,axiom,
! [A: set_a,C2: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_1024_Un__mono,axiom,
! [A: set_list_a,C2: set_list_a,B2: set_list_a,D: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ D )
=> ( ord_le8861187494160871172list_a @ ( sup_sup_set_list_a @ A @ B2 ) @ ( sup_sup_set_list_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_1025_Un__empty__left,axiom,
! [B2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1026_Un__empty__left,axiom,
! [B2: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1027_Un__empty__left,axiom,
! [B2: set_set_list_a] :
( ( sup_su4537662296134749976list_a @ bot_bo3186585308812441520list_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1028_Un__empty__left,axiom,
! [B2: set_set_a] :
( ( sup_sup_set_set_a @ bot_bot_set_set_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_1029_Un__empty__right,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% Un_empty_right
thf(fact_1030_Un__empty__right,axiom,
! [A: set_list_a] :
( ( sup_sup_set_list_a @ A @ bot_bot_set_list_a )
= A ) ).
% Un_empty_right
thf(fact_1031_Un__empty__right,axiom,
! [A: set_set_list_a] :
( ( sup_su4537662296134749976list_a @ A @ bot_bo3186585308812441520list_a )
= A ) ).
% Un_empty_right
thf(fact_1032_Un__empty__right,axiom,
! [A: set_set_a] :
( ( sup_sup_set_set_a @ A @ bot_bot_set_set_a )
= A ) ).
% Un_empty_right
thf(fact_1033_Diff__Int__distrib2,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A @ B2 ) @ C2 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1034_Diff__Int__distrib2,axiom,
! [A: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( inf_inf_set_list_a @ ( minus_646659088055828811list_a @ A @ B2 ) @ C2 )
= ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ A @ C2 ) @ ( inf_inf_set_list_a @ B2 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1035_Diff__Int__distrib,axiom,
! [C2: set_a,A: set_a,B2: set_a] :
( ( inf_inf_set_a @ C2 @ ( minus_minus_set_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C2 @ A ) @ ( inf_inf_set_a @ C2 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_1036_Diff__Int__distrib,axiom,
! [C2: set_list_a,A: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ C2 @ ( minus_646659088055828811list_a @ A @ B2 ) )
= ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ C2 @ A ) @ ( inf_inf_set_list_a @ C2 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_1037_Diff__Diff__Int,axiom,
! [A: set_a,B2: set_a] :
( ( minus_minus_set_a @ A @ ( minus_minus_set_a @ A @ B2 ) )
= ( inf_inf_set_a @ A @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_1038_Diff__Diff__Int,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ A @ ( minus_646659088055828811list_a @ A @ B2 ) )
= ( inf_inf_set_list_a @ A @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_1039_Diff__Int2,axiom,
! [A: set_a,C2: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A @ C2 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_1040_Diff__Int2,axiom,
! [A: set_list_a,C2: set_list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ A @ C2 ) @ ( inf_inf_set_list_a @ B2 @ C2 ) )
= ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ A @ C2 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_1041_Int__Diff,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_a @ A @ ( minus_minus_set_a @ B2 @ C2 ) ) ) ).
% Int_Diff
thf(fact_1042_Int__Diff,axiom,
! [A: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_list_a @ A @ ( minus_646659088055828811list_a @ B2 @ C2 ) ) ) ).
% Int_Diff
thf(fact_1043_Un__Int__distrib2,axiom,
! [B2: set_a,C2: set_a,A: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B2 @ C2 ) @ A )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B2 @ A ) @ ( sup_sup_set_a @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_1044_x_Oideal__is__subalgebra,axiom,
! [K: set_list_a,I3: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ideal_8896367198367571637t_unit @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( embedd1768981623711841426t_unit @ K @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.ideal_is_subalgebra
thf(fact_1045_x_Oa__lcos__m__assoc,axiom,
! [M: set_list_a,G: list_a,H: list_a] :
( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ 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 ) ) @ G @ H ) @ M ) ) ) ) ) ).
% x.a_lcos_m_assoc
thf(fact_1046_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_1047_x_Oup__one__closed,axiom,
( member_nat_list_a
@ ^ [N4: nat] : ( if_list_a @ ( N4 = zero_zero_nat ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.up_one_closed
thf(fact_1048_x_Oup__minus__closed,axiom,
! [P: nat > list_a,Q: nat > list_a] :
( ( member_nat_list_a @ P @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_nat_list_a @ Q @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_nat_list_a
@ ^ [I: nat] : ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( P @ I ) @ ( Q @ I ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.up_minus_closed
thf(fact_1049_x_Osubalgebra__inter,axiom,
! [K: set_list_a,V: set_list_a,V2: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1768981623711841426t_unit @ K @ V2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( embedd1768981623711841426t_unit @ K @ ( inf_inf_set_list_a @ V @ V2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subalgebra_inter
thf(fact_1050_up__one__closed,axiom,
( member_nat_a
@ ^ [N4: nat] : ( if_a @ ( N4 = zero_zero_nat ) @ ( one_a_ring_ext_a_b @ r ) @ ( zero_a_b @ r ) )
@ ( up_a_b @ r ) ) ).
% up_one_closed
thf(fact_1051_x_Oa__l__coset__subset__G,axiom,
! [H2: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ H2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.a_l_coset_subset_G
thf(fact_1052_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_1053_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_1054_x_Ominus__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.minus_closed
thf(fact_1055_x_Or__right__minus__eq,axiom,
! [A2: list_a,B3: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( A2 = B3 ) ) ) ) ).
% x.r_right_minus_eq
thf(fact_1056_euclidean__domain__axioms,axiom,
( ring_e8745995371659049232in_a_b @ r
@ ^ [Uu: a] : zero_zero_nat ) ).
% euclidean_domain_axioms
thf(fact_1057_x_Ohom__sub,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ x )
= ( a_minus_a_b @ r @ ( eval_a_b @ r @ X @ x ) @ ( eval_a_b @ r @ Y @ x ) ) ) ) ) ).
% x.hom_sub
thf(fact_1058_x_Oring_Oimg__is__subalgebra,axiom,
! [K: set_list_a,V: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( embedd9027525575939734154ra_a_b
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ K )
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ V )
@ r ) ) ) ).
% x.ring.img_is_subalgebra
thf(fact_1059_x_Ocgenideal__eq__genideal,axiom,
! [I4: list_a] :
( ( member_list_a @ I4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 )
= ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I4 @ bot_bot_set_list_a ) ) ) ) ).
% x.cgenideal_eq_genideal
thf(fact_1060_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_1061_x_Ozeroideal,axiom,
ideal_8896367198367571637t_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.zeroideal
thf(fact_1062_x_Ogenideal__self_H,axiom,
! [I4: list_a] :
( ( member_list_a @ I4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ I4 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I4 @ bot_bot_set_list_a ) ) ) ) ).
% x.genideal_self'
thf(fact_1063_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_1064_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_1065_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_1066_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_1067_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_1068_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_1069_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_1070_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_1071_x_OIdl__subset__ideal_H,axiom,
! [A2: list_a,B3: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( 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 @ B3 @ bot_bot_set_list_a ) ) )
= ( member_list_a @ A2 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B3 @ bot_bot_set_list_a ) ) ) ) ) ) ).
% x.Idl_subset_ideal'
thf(fact_1072_x_Oring_Oline__extension__hom,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 ) ) ) )
=> ( ( embedd971793762689825387on_a_b @ r
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ K )
@ ( eval_a_b @ r @ A2 @ x )
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ E ) )
= ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A2 @ E ) ) ) ) ) ) ).
% x.ring.line_extension_hom
thf(fact_1073_x_Otrivialideals__eq__field,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 ) )
=> ( ( ( collect_set_list_a
@ ^ [I6: set_list_a] : ( ideal_8896367198367571637t_unit @ I6 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( insert_set_list_a @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( insert_set_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bo3186585308812441520list_a ) ) )
= ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.trivialideals_eq_field
thf(fact_1074_x_Otrivialideals__fieldI,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 ) )
=> ( ( ( collect_set_list_a
@ ^ [I6: set_list_a] : ( ideal_8896367198367571637t_unit @ I6 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( insert_set_list_a @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( insert_set_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bo3186585308812441520list_a ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.trivialideals_fieldI
thf(fact_1075_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 ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( 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 @ X2 @ ( 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_1076_add_Osurj__const__mult,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( image_a_a @ ( add_a_b @ r @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) )
= ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.surj_const_mult
thf(fact_1077_zeroideal,axiom,
ideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroideal
thf(fact_1078_genideal__self_H,axiom,
! [I4: a] :
( ( member_a @ I4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I4 @ ( genideal_a_b @ r @ ( insert_a @ I4 @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_1079_genideal__zero,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% genideal_zero
thf(fact_1080_zeromaximalideal,axiom,
maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeromaximalideal
thf(fact_1081_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_1082_x_OUnits__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_closed
thf(fact_1083_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_1084_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_1085_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_1086_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_1087_x_Oadd_Osurj__const__mult,axiom,
! [A2: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( image_list_a_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.surj_const_mult
thf(fact_1088_Idl__subset__ideal_H,axiom,
! [A2: a,B3: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B3 @ bot_bot_set_a ) ) )
= ( member_a @ A2 @ ( genideal_a_b @ r @ ( insert_a @ B3 @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_1089_genideal__one,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) )
= ( partia707051561876973205xt_a_b @ r ) ) ).
% genideal_one
thf(fact_1090_all__ideals,axiom,
( ( collect_set_a
@ ^ [I6: set_a] : ( ideal_a_b @ I6 @ r ) )
= ( insert_set_a @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ ( insert_set_a @ ( partia707051561876973205xt_a_b @ r ) @ bot_bot_set_set_a ) ) ) ).
% all_ideals
thf(fact_1091_cgenideal__eq__genideal,axiom,
! [I4: a] :
( ( member_a @ I4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( cgenid547466209912283029xt_a_b @ r @ I4 )
= ( genideal_a_b @ r @ ( insert_a @ I4 @ bot_bot_set_a ) ) ) ) ).
% cgenideal_eq_genideal
thf(fact_1092_x_Oprod__unit__l,axiom,
! [A2: list_a,B3: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B3 ) @ ( 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 @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ B3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.prod_unit_l
thf(fact_1093_x_Oprod__unit__r,axiom,
! [A2: list_a,B3: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B3 ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( 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 @ B3 @ ( 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_1094_x_Ounit__factor,axiom,
! [A2: list_a,B3: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B3 ) @ ( 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 @ B3 @ ( 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_1095_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_1096_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_1097_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_1098_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_1099_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_1100_x_OUnits__inv__comm,axiom,
! [X: list_a,Y: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( 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 @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.Units_inv_comm
thf(fact_1101_x_Oideal__eq__carrier__iff,axiom,
! [A2: list_a] :
( ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 ) )
= ( member_list_a @ A2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.ideal_eq_carrier_iff
thf(fact_1102_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 ) )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ K )
& ? [Y3: list_a] :
( ( member_list_a @ Y3 @ 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 ) ) @ X3 @ A2 ) @ Y3 ) ) ) ) ) ) ).
% x.line_extension_mem_iff
thf(fact_1103_trivialideals__eq__field,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( ( ( collect_set_a
@ ^ [I6: set_a] : ( ideal_a_b @ I6 @ r ) )
= ( insert_set_a @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ ( insert_set_a @ ( partia707051561876973205xt_a_b @ r ) @ bot_bot_set_set_a ) ) )
= ( field_a_b @ r ) ) ) ).
% trivialideals_eq_field
thf(fact_1104_trivialideals__fieldI,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( ( ( collect_set_a
@ ^ [I6: set_a] : ( ideal_a_b @ I6 @ r ) )
= ( insert_set_a @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ ( insert_set_a @ ( partia707051561876973205xt_a_b @ r ) @ bot_bot_set_set_a ) ) )
=> ( field_a_b @ r ) ) ) ).
% trivialideals_fieldI
thf(fact_1105_x_OUnits__r__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Units_r_inv_ex
thf(fact_1106_x_OUnits__l__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Units_l_inv_ex
thf(fact_1107_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_1108_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_1109_mult__divides,axiom,
! [A2: a,B3: a,C: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( 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 @ B3 ) )
=> ( factor8216151070175719842xt_a_b @ r @ A2 @ B3 ) ) ) ) ) ).
% mult_divides
thf(fact_1110_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_1111_x_OUnits__m__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( 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 ) ) @ X @ Y ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Units_m_closed
thf(fact_1112_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_1113_x_OUnits__l__cancel,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( 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 ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% x.Units_l_cancel
thf(fact_1114_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_1115_primeideal__iff__prime,axiom,
! [P: a] :
( ( member_a @ P @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P ) @ r )
= ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% primeideal_iff_prime
thf(fact_1116_field__iff__prime,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( field_6045675692312731021t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A2 ) ) )
= ( ring_ring_prime_a_b @ r @ A2 ) ) ) ).
% field_iff_prime
thf(fact_1117_euclidean__domainI,axiom,
! [Phi: a > nat] :
( ! [A6: a,B5: a] :
( ( member_a @ A6 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B5 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ? [Q3: a,R6: a] :
( ( member_a @ Q3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ R6 @ ( partia707051561876973205xt_a_b @ r ) )
& ( A6
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ B5 @ Q3 ) @ R6 ) )
& ( ( R6
= ( zero_a_b @ r ) )
| ( ord_less_nat @ ( Phi @ R6 ) @ ( Phi @ B5 ) ) ) ) ) )
=> ( ring_e8745995371659049232in_a_b @ r @ Phi ) ) ).
% euclidean_domainI
thf(fact_1118_Units__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_1119_prod__unit__l,axiom,
! [A2: a,B3: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A2 @ B3 ) @ ( 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 @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_1120_prod__unit__r,axiom,
! [A2: a,B3: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A2 @ B3 ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_1121_unit__factor,axiom,
! [A2: a,B3: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A2 @ B3 ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_1122_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_1123_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_1124_Units__inv__comm,axiom,
! [X: a,Y: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( 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 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_inv_comm
thf(fact_1125_ideal__eq__carrier__iff,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( partia707051561876973205xt_a_b @ r )
= ( cgenid547466209912283029xt_a_b @ r @ A2 ) )
= ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ideal_eq_carrier_iff
thf(fact_1126_ring__irreducibleE_I4_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_1127_Units__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X2 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_1128_Units__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_1129_maximalideal__prime,axiom,
! [I3: set_a] :
( ( maximalideal_a_b @ I3 @ r )
=> ( primeideal_a_b @ I3 @ r ) ) ).
% maximalideal_prime
thf(fact_1130_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_1131_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_1132_ring__irreducibleE_I5_J,axiom,
! [R2: a,A2: a,B3: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ r @ A2 @ B3 ) )
=> ( ( member_a @ A2 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_1133_zeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroprimeideal
thf(fact_1134_boundD__carrier,axiom,
! [N: nat,F: nat > a,M2: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F )
=> ( ( ord_less_nat @ N @ M2 )
=> ( member_a @ ( F @ M2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_1135_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_1136_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_1137_x_OboundD__carrier,axiom,
! [N: nat,F: nat > list_a,M2: nat] :
( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N @ F )
=> ( ( ord_less_nat @ N @ M2 )
=> ( member_list_a @ ( F @ M2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.boundD_carrier
thf(fact_1138_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_1139_ring__irreducibleI,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ! [A6: a,B5: a] :
( ( member_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ r @ A6 @ B5 ) )
=> ( ( member_a @ A6 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B5 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ r @ R2 ) ) ) ) ).
% ring_irreducibleI
thf(fact_1140_field__intro2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) ) )
=> ( field_a_b @ r ) ) ) ).
% field_intro2
thf(fact_1141_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 ) )
=> ~ ! [B5: a] :
( ( member_a @ B5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ B5 )
=> ~ ( factor8216151070175719842xt_a_b @ r @ B5 @ A2 ) ) ) ) ) ).
% exists_irreducible_divisor
thf(fact_1142_Units__m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( 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 @ X @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_1143_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_1144_euclidean__function,axiom,
! [A2: a,B3: a] :
( ( member_a @ A2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ? [Q4: a,R7: a] :
( ( member_a @ Q4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ R7 @ ( partia707051561876973205xt_a_b @ r ) )
& ( A2
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ B3 @ Q4 ) @ R7 ) )
& ( ( R7
= ( zero_a_b @ r ) )
| ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ) ) ).
% euclidean_function
thf(fact_1145_Units__l__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( 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 @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% Units_l_cancel
thf(fact_1146_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_1147_FactRing__zeroideal_I2_J,axiom,
is_rin9099215527551818550t_unit @ r @ ( factRing_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% FactRing_zeroideal(2)
thf(fact_1148_FactRing__zeroideal_I1_J,axiom,
is_rin6001486760346555702it_a_b @ ( factRing_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) @ r ).
% FactRing_zeroideal(1)
thf(fact_1149_x_Oorder__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_3240872107759947550t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.order_gt_0_iff_finite
thf(fact_1150_order__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_a_ring_ext_a_b @ r ) )
= ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% order_gt_0_iff_finite
thf(fact_1151_alg__mult__gt__zero__iff__is__root,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P @ X ) )
= ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_1152_x_Omaximalideal__prime,axiom,
! [I3: set_list_a] :
( ( maxima6585700282301356660t_unit @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( primei6309817859076077608t_unit @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.maximalideal_prime
thf(fact_1153_x_OFactRing__zeroideal_I2_J,axiom,
is_rin2993610189962786360t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ).
% x.FactRing_zeroideal(2)
thf(fact_1154_x_OFactRing__zeroideal_I1_J,axiom,
is_rin4843644836746533432t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.FactRing_zeroideal(1)
thf(fact_1155_x_Oring_Onon__trivial__field__hom__imp__inj,axiom,
( ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.ring.non_trivial_field_hom_imp_inj
thf(fact_1156_domain__iff__prime,axiom,
! [A2: a] :
( ( member_a @ A2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A2 ) ) )
= ( ring_ring_prime_a_b @ r @ A2 ) ) ) ).
% domain_iff_prime
thf(fact_1157_x_Oring_Otrivial__hom__iff,axiom,
( ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
= ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.ring.trivial_hom_iff
thf(fact_1158_quot__domain__iff__primeideal,axiom,
! [P2: set_a] :
( ( ideal_a_b @ P2 @ r )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ P2 ) )
= ( primeideal_a_b @ P2 @ r ) ) ) ).
% quot_domain_iff_primeideal
thf(fact_1159_quot__domain__imp__primeideal,axiom,
! [P2: set_a] :
( ( ideal_a_b @ P2 @ r )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ P2 ) )
=> ( primeideal_a_b @ P2 @ r ) ) ) ).
% quot_domain_imp_primeideal
thf(fact_1160_x_Oring_Okernel__is__ideal,axiom,
( ideal_8896367198367571637t_unit
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.ring.kernel_is_ideal
thf(fact_1161_x_Oring_Othe__elem__wf,axiom,
! [X5: set_list_a] :
( ( member_set_list_a @ X5
@ ( partia141011252114345353t_unit
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X5 )
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% x.ring.the_elem_wf
thf(fact_1162_x_Oring_Oinj__iff__trivial__ker,axiom,
( ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ).
% x.ring.inj_iff_trivial_ker
thf(fact_1163_x_Oring_Otrivial__ker__imp__inj,axiom,
( ( ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
=> ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.ring.trivial_ker_imp_inj
thf(fact_1164_x_Oring_Othe__elem__wf_H,axiom,
! [X5: set_list_a] :
( ( member_set_list_a @ X5
@ ( partia141011252114345353t_unit
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X5 )
= ( insert_a @ ( eval_a_b @ r @ X2 @ x ) @ bot_bot_set_a ) ) ) ) ).
% x.ring.the_elem_wf'
thf(fact_1165_x_Oring_OFactRing__iso,axiom,
( ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( partia707051561876973205xt_a_b @ r ) )
=> ( is_rin5597148638330396976it_a_b
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) )
@ r ) ) ).
% x.ring.FactRing_iso
thf(fact_1166_x_Oring_OA__FactGroup__nonempty,axiom,
! [X5: set_list_a] :
( ( member_set_list_a @ X5
@ ( partia5178357399839081912t_unit
@ ( a_Fact452226231247776317t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
=> ( X5 != bot_bot_set_list_a ) ) ).
% x.ring.A_FactGroup_nonempty
thf(fact_1167_x_Oring_Othe__elem__surj,axiom,
( ( image_set_list_a_a
@ ^ [X6: set_list_a] :
( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X6 ) )
@ ( partia141011252114345353t_unit
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
= ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.ring.the_elem_surj
thf(fact_1168_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_1169_add_Oinj__on__multc,axiom,
! [C: a] :
( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( inj_on_a_a
@ ^ [X3: a] : ( add_a_b @ r @ X3 @ C )
@ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.inj_on_multc
thf(fact_1170_add_Oinj__on__cmult,axiom,
! [C: a] :
( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( inj_on_a_a @ ( add_a_b @ r @ C ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.inj_on_cmult
thf(fact_1171_x_Oeval__in__carrier__2,axiom,
! [X: list_list_a,Y: list_a] :
( ( member_list_list_a @ X @ ( 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 ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.eval_in_carrier_2
thf(fact_1172_add_Oinj__on__g,axiom,
! [H2: set_a,A2: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( inj_on_a_a
@ ^ [Y3: a] : ( add_a_b @ r @ Y3 @ A2 )
@ H2 ) ) ) ).
% add.inj_on_g
thf(fact_1173_x_Oadd_Oinj__on__multc,axiom,
! [C: list_a] :
( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( inj_on_list_a_list_a
@ ^ [X3: list_a] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ C )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.inj_on_multc
thf(fact_1174_x_Oadd_Oinj__on__cmult,axiom,
! [C: list_a] :
( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( inj_on_list_a_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.inj_on_cmult
thf(fact_1175_domain__eq__zeroprimeideal,axiom,
( ( domain_a_b @ r )
= ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ) ) ).
% domain_eq_zeroprimeideal
thf(fact_1176_zeroprimeideal__domainI,axiom,
( ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
=> ( domain_a_b @ r ) ) ).
% zeroprimeideal_domainI
thf(fact_1177_x_Oadd_Oinj__on__g,axiom,
! [H2: set_list_a,A2: list_a] :
( ( ord_le8861187494160871172list_a @ H2 @ ( 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 ) ) ) )
=> ( inj_on_list_a_list_a
@ ^ [Y3: list_a] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y3 @ A2 )
@ H2 ) ) ) ).
% x.add.inj_on_g
thf(fact_1178_x_Oquot__domain__iff__primeideal,axiom,
! [P2: set_list_a] :
( ( ideal_8896367198367571637t_unit @ P2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( domain1617769409708967785t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) )
= ( primei6309817859076077608t_unit @ P2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.quot_domain_iff_primeideal
thf(fact_1179_x_Oquot__domain__imp__primeideal,axiom,
! [P2: set_list_a] :
( ( ideal_8896367198367571637t_unit @ P2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( domain1617769409708967785t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) )
=> ( primei6309817859076077608t_unit @ P2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.quot_domain_imp_primeideal
thf(fact_1180_x_Ozeroprimeideal__domainI,axiom,
( ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.zeroprimeideal_domainI
thf(fact_1181_x_Odomain__eq__zeroprimeideal,axiom,
( ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.domain_eq_zeroprimeideal
thf(fact_1182_x_Oring_Oinj__on__domain,axiom,
( ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( domain_a_b @ r )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.ring.inj_on_domain
thf(fact_1183_lagrange__basis__polynomial__aux__def,axiom,
! [S: set_a] :
( ( lagran9092808442999052491ux_a_b @ r @ S )
= ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [S3: a] : ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ ( poly_of_const_a_b @ r @ S3 ) )
@ S ) ) ).
% lagrange_basis_polynomial_aux_def
thf(fact_1184_x_Oring_Othe__elem__inj,axiom,
! [X5: set_list_a,Y6: set_list_a] :
( ( member_set_list_a @ X5
@ ( partia141011252114345353t_unit
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
=> ( ( member_set_list_a @ Y6
@ ( partia141011252114345353t_unit
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
=> ( ( ( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X5 ) )
= ( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ Y6 ) ) )
=> ( X5 = Y6 ) ) ) ) ).
% x.ring.the_elem_inj
thf(fact_1185_x_Oring_OA__FactGroup__inj__on,axiom,
( inj_on_set_list_a_a
@ ^ [X6: set_list_a] :
( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X6 ) )
@ ( partia5178357399839081912t_unit
@ ( a_Fact452226231247776317t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) ) ).
% x.ring.A_FactGroup_inj_on
thf(fact_1186_x_Oring_OFactGroup__the__elem__mem,axiom,
! [X5: set_list_a] :
( ( member_set_list_a @ X5
@ ( partia5178357399839081912t_unit
@ ( a_Fact452226231247776317t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
=> ( member_a
@ ( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X5 ) )
@ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.ring.FactGroup_the_elem_mem
thf(fact_1187_x_Oring_OA__FactGroup__onto,axiom,
( ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( partia707051561876973205xt_a_b @ r ) )
=> ( ( image_set_list_a_a
@ ^ [X6: set_list_a] :
( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X6 ) )
@ ( partia5178357399839081912t_unit
@ ( a_Fact452226231247776317t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
= ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.ring.A_FactGroup_onto
thf(fact_1188_x_Oring_OFactRing__iso__set,axiom,
( ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( partia707051561876973205xt_a_b @ r ) )
=> ( member_set_list_a_a
@ ^ [X6: set_list_a] :
( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X6 ) )
@ ( ring_i8122894263081988538it_a_b
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) )
@ r ) ) ) ).
% x.ring.FactRing_iso_set
thf(fact_1189_x_Oring_Othe__elem__hom,axiom,
( member_set_list_a_a
@ ^ [X6: set_list_a] :
( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X6 ) )
@ ( ring_h8906680420194085028it_a_b
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) )
@ r ) ) ).
% x.ring.the_elem_hom
thf(fact_1190_x_Oeval__var,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( 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 ) ) ) @ X )
= X ) ) ).
% x.eval_var
thf(fact_1191_x_Oring_Oadditive__subgroup__a__kernel,axiom,
( additi4714453376129182166t_unit
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.ring.additive_subgroup_a_kernel
thf(fact_1192_x_Oring_Ohomh,axiom,
( member_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( ring_h2895973938487309444it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r ) ) ).
% x.ring.homh
thf(fact_1193_x_Olagrange__basis__polynomial__aux__def,axiom,
! [S: set_list_a] :
( ( lagran3534788790333317459t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S )
= ( finpro3417560807142560175list_a @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ ^ [S3: list_a] : ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S3 ) )
@ S ) ) ).
% x.lagrange_basis_polynomial_aux_def
thf(fact_1194_x_Oeval__poly__of__const,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( 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 ) ) @ X ) @ Y )
= X ) ) ).
% x.eval_poly_of_const
thf(fact_1195_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_1196_x_Oring_Oset__add__ker__hom_I2_J,axiom,
! [I3: set_list_a] :
( ( ord_le8861187494160871172list_a @ I3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
@ I3 ) )
= ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ I3 ) ) ) ).
% x.ring.set_add_ker_hom(2)
thf(fact_1197_x_Oadd__ideals,axiom,
! [I3: set_list_a,J2: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ideal_8896367198367571637t_unit @ J2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ideal_8896367198367571637t_unit @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I3 @ J2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add_ideals
thf(fact_1198_x_Oadd__additive__subgroups,axiom,
! [H2: set_list_a,K: set_list_a] :
( ( additi4714453376129182166t_unit @ H2 @ ( 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 ) ) @ H2 @ K ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add_additive_subgroups
thf(fact_1199_x_Oset__add__closed,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.set_add_closed
thf(fact_1200_x_Oset__add__comm,axiom,
! [I3: set_list_a,J2: set_list_a] :
( ( ord_le8861187494160871172list_a @ I3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ J2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I3 @ J2 )
= ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ J2 @ I3 ) ) ) ) ).
% x.set_add_comm
thf(fact_1201_x_Osetadd__subset__G,axiom,
! [H2: set_list_a,K: set_list_a] :
( ( ord_le8861187494160871172list_a @ H2 @ ( 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 ) ) @ H2 @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.setadd_subset_G
thf(fact_1202_x_Ounion__genideal,axiom,
! [I3: set_list_a,J2: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ideal_8896367198367571637t_unit @ J2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( sup_sup_set_list_a @ I3 @ J2 ) )
= ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I3 @ J2 ) ) ) ) ).
% x.union_genideal
thf(fact_1203_x_Oring_Oset__add__ker__hom_I1_J,axiom,
! [I3: set_list_a] :
( ( ord_le8861187494160871172list_a @ I3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I3
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) )
= ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ I3 ) ) ) ).
% x.ring.set_add_ker_hom(1)
thf(fact_1204_x_Oring_OFactRing__iso__set__aux,axiom,
( member_set_list_a_a
@ ^ [X6: set_list_a] :
( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X6 ) )
@ ( ring_i8122894263081988538it_a_b
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] :
( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ r ) ) ) ).
% x.ring.FactRing_iso_set_aux
thf(fact_1205_x_Oring_Oabelian__subgroup__a__kernel,axiom,
( abelia6695205329122750356t_unit
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.ring.abelian_subgroup_a_kernel
thf(fact_1206_add__additive__subgroups,axiom,
! [H2: set_a,K: set_a] :
( ( additi2834746164131130830up_a_b @ H2 @ r )
=> ( ( additi2834746164131130830up_a_b @ K @ r )
=> ( additi2834746164131130830up_a_b @ ( set_add_a_b @ r @ H2 @ K ) @ r ) ) ) ).
% add_additive_subgroups
thf(fact_1207_add__ideals,axiom,
! [I3: set_a,J2: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ( ( ideal_a_b @ J2 @ r )
=> ( ideal_a_b @ ( set_add_a_b @ r @ I3 @ J2 ) @ r ) ) ) ).
% add_ideals
thf(fact_1208_set__add__closed,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A @ B2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_add_closed
thf(fact_1209_set__add__comm,axiom,
! [I3: set_a,J2: set_a] :
( ( ord_less_eq_set_a @ I3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ J2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ I3 @ J2 )
= ( set_add_a_b @ r @ J2 @ I3 ) ) ) ) ).
% set_add_comm
thf(fact_1210_setadd__subset__G,axiom,
! [H2: set_a,K: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H2 @ K ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% setadd_subset_G
thf(fact_1211_union__genideal,axiom,
! [I3: set_a,J2: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ( ( ideal_a_b @ J2 @ r )
=> ( ( genideal_a_b @ r @ ( sup_sup_set_a @ I3 @ J2 ) )
= ( set_add_a_b @ r @ I3 @ J2 ) ) ) ) ).
% union_genideal
thf(fact_1212_ideal__sum__iff__gcd,axiom,
! [A2: a,B3: a,D2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ D2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( cgenid547466209912283029xt_a_b @ r @ D2 )
= ( set_add_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A2 ) @ ( cgenid547466209912283029xt_a_b @ r @ B3 ) ) )
= ( isgcd_a_ring_ext_a_b @ r @ D2 @ A2 @ B3 ) ) ) ) ) ).
% ideal_sum_iff_gcd
thf(fact_1213_x_Oring_Oimg__is__domain,axiom,
( ( domain_a_b @ r )
=> ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] :
( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ r ) ) ) ).
% x.ring.img_is_domain
thf(fact_1214_subdomain__iff,axiom,
! [H2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( subdomain_a_b @ H2 @ r )
= ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H2
@ r ) ) ) ) ).
% subdomain_iff
thf(fact_1215_subdomain__is__domain,axiom,
! [H2: set_a] :
( ( subdomain_a_b @ H2 @ r )
=> ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H2
@ r ) ) ) ).
% subdomain_is_domain
thf(fact_1216_bezout__identity,axiom,
! [A2: a,B3: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A2 ) @ ( cgenid547466209912283029xt_a_b @ r @ B3 ) )
= ( cgenid547466209912283029xt_a_b @ r @ ( somegc1600592057159103747xt_a_b @ r @ A2 @ B3 ) ) ) ) ) ).
% bezout_identity
thf(fact_1217_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_1218_x_Oring__axioms,axiom,
ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.ring_axioms
thf(fact_1219_x_Oring_Oimg__is__ring,axiom,
( ring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] :
( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ r ) ) ).
% x.ring.img_is_ring
thf(fact_1220_x_Osubdomain__iff,axiom,
! [H2: set_list_a] :
( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( subdom7821232466298058046t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( domain6553523120543210313t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H2
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.subdomain_iff
thf(fact_1221_x_Osubdomain__is__domain,axiom,
! [H2: set_list_a] :
( ( subdom7821232466298058046t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( domain6553523120543210313t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H2
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subdomain_is_domain
thf(fact_1222_x_Oexp__base__closed,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( polyno3522816881121920896t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.exp_base_closed
thf(fact_1223_x_Oeval__in__carrier,axiom,
! [P: list_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.eval_in_carrier
thf(fact_1224_x_OsubdomainI,axiom,
! [H2: set_list_a] :
( ( subcri7763218559781929323t_unit @ H2 @ ( 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 ) ) ) )
=> ( ! [H1: list_a,H22: list_a] :
( ( member_list_a @ H1 @ H2 )
=> ( ( member_list_a @ H22 @ H2 )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( H1
= ( 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 @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.subdomainI
thf(fact_1225_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_1226_x_Osubcring__inter,axiom,
! [I3: set_list_a,J2: set_list_a] :
( ( subcri7763218559781929323t_unit @ I3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subcri7763218559781929323t_unit @ J2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subcri7763218559781929323t_unit @ ( inf_inf_set_list_a @ I3 @ J2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subcring_inter
thf(fact_1227_x_OsubfieldI,axiom,
! [K: set_list_a] :
( ( subcri7763218559781929323t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( units_2932844235741507942t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subfieldI
thf(fact_1228_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_1229_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_1230_x_Osubring__props_I7_J,axiom,
! [K: set_list_a,H12: list_a,H23: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H12 @ K )
=> ( ( member_list_a @ H23 @ K )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H12 @ H23 ) @ K ) ) ) ) ).
% x.subring_props(7)
thf(fact_1231_x_Osubring__props_I6_J,axiom,
! [K: set_list_a,H12: list_a,H23: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H12 @ K )
=> ( ( member_list_a @ H23 @ K )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H12 @ H23 ) @ K ) ) ) ) ).
% x.subring_props(6)
thf(fact_1232_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_1233_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_1234_univ__poly__carrier__subfield__of__consts,axiom,
subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% univ_poly_carrier_subfield_of_consts
thf(fact_1235_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_1236_x_Osubfield__iff_I2_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( field_6388047844668329575t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subfield_iff(2)
thf(fact_1237_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_1238_x_Oline__extension__smult__closed,axiom,
! [K: set_list_a,E: set_list_a,A2: list_a,K2: list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [K3: list_a,V4: list_a] :
( ( member_list_a @ K3 @ K )
=> ( ( member_list_a @ V4 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ V4 ) @ 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 @ K2 @ 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 ) ) @ K2 @ U ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A2 @ E ) ) ) ) ) ) ) ) ).
% x.line_extension_smult_closed
thf(fact_1239_x_Osubfield__iff_I1_J,axiom,
! [K: set_list_a] :
( ( field_6388047844668329575t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : 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 ) ) ) )
=> ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subfield_iff(1)
thf(fact_1240_x_Osubfield__m__inv__simprule,axiom,
! [K: set_list_a,K2: list_a,A2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ 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 ) ) @ K2 @ A2 ) @ K )
=> ( member_list_a @ A2 @ K ) ) ) ) ) ).
% x.subfield_m_inv_simprule
thf(fact_1241_x_Oring_Oimg__is__subfield_I1_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ K ) ) ) ).
% x.ring.img_is_subfield(1)
thf(fact_1242_x_Oring_Oinj__on__Span__iff__trivial__ker,axiom,
! [K: set_list_a,Us: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) )
= ( ( a_kern7116238624728830086it_a_b
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us )
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
@ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ) ).
% x.ring.inj_on_Span_iff_trivial_ker
thf(fact_1243_carrier__is__subfield,axiom,
subfield_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subfield
thf(fact_1244_carrier__is__subcring,axiom,
subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subcring
thf(fact_1245_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_1246_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_1247_subring__props_I7_J,axiom,
! [K: set_a,H12: a,H23: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H12 @ K )
=> ( ( member_a @ H23 @ K )
=> ( member_a @ ( add_a_b @ r @ H12 @ H23 ) @ K ) ) ) ) ).
% subring_props(7)
thf(fact_1248_subring__props_I6_J,axiom,
! [K: set_a,H12: a,H23: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H12 @ K )
=> ( ( member_a @ H23 @ K )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H23 ) @ K ) ) ) ) ).
% subring_props(6)
thf(fact_1249_subring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( K != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_1250_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_1251_subcring__inter,axiom,
! [I3: set_a,J2: set_a] :
( ( subcring_a_b @ I3 @ r )
=> ( ( subcring_a_b @ J2 @ r )
=> ( subcring_a_b @ ( inf_inf_set_a @ I3 @ J2 ) @ r ) ) ) ).
% subcring_inter
thf(fact_1252_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_1253_subfield__iff_I2_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( field_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) ) ) ).
% subfield_iff(2)
thf(fact_1254_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_1255_line__extension__smult__closed,axiom,
! [K: set_a,E: set_a,A2: a,K2: a,U: a] :
( ( subfield_a_b @ K @ r )
=> ( ! [K3: a,V4: a] :
( ( member_a @ K3 @ K )
=> ( ( member_a @ V4 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ V4 ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A2 @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ U ) @ ( embedd971793762689825387on_a_b @ r @ K @ A2 @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_1256_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_1257_univ__poly__subfield__of__consts,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_subfield_of_consts
thf(fact_1258_subdomainI,axiom,
! [H2: set_a] :
( ( subcring_a_b @ H2 @ r )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( ( ( mult_a_ring_ext_a_b @ r @ H1 @ H22 )
= ( zero_a_b @ r ) )
=> ( ( H1
= ( zero_a_b @ r ) )
| ( H22
= ( zero_a_b @ r ) ) ) ) ) )
=> ( subdomain_a_b @ H2 @ r ) ) ) ) ).
% subdomainI
thf(fact_1259_subfield__iff_I1_J,axiom,
! [K: set_a] :
( ( field_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( subfield_a_b @ K @ r ) ) ) ).
% subfield_iff(1)
thf(fact_1260_x_Ouniv__poly__subfield__of__consts,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subfie4546268998243038636t_unit @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ).
% x.univ_poly_subfield_of_consts
thf(fact_1261_x_OSpan__in__carrier,axiom,
! [K: set_list_a,Us: list_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( 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 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Span_in_carrier
thf(fact_1262_subfield__m__inv__simprule,axiom,
! [K: set_a,K2: a,A2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ A2 ) @ K )
=> ( member_a @ A2 @ K ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_1263_subfieldI,axiom,
! [K: set_a] :
( ( subcring_a_b @ K @ r )
=> ( ( ( units_a_ring_ext_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) )
= ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( subfield_a_b @ K @ r ) ) ) ).
% subfieldI
thf(fact_1264_x_OSpan__strict__incl,axiom,
! [K: set_list_a,Us: 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 @ Us ) @ ( 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 @ Us ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Vs ) )
& ~ ( member_list_a @ X2 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) ) ) ) ) ) ) ).
% x.Span_strict_incl
thf(fact_1265_x_OSpan__subgroup__props_I1_J,axiom,
! [K: set_list_a,Us: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( 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 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Span_subgroup_props(1)
thf(fact_1266_x_OSpan__base__incl,axiom,
! [K: set_list_a,Us: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) ) ) ) ).
% x.Span_base_incl
thf(fact_1267_x_OSpan__same__set,axiom,
! [K: set_list_a,Us: 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 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( set_list_a2 @ Us )
= ( set_list_a2 @ Vs ) )
=> ( ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us )
= ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) ) ) ) ) ).
% x.Span_same_set
thf(fact_1268_x_Omono__Span__sublist,axiom,
! [K: set_list_a,Us: 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 @ Us ) @ ( 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 @ Us ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) ) ) ) ) ).
% x.mono_Span_sublist
thf(fact_1269_x_Omono__Span__subset,axiom,
! [K: set_list_a,Us: 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 @ Us ) @ ( 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 @ Us ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) ) ) ) ) ).
% x.mono_Span_subset
thf(fact_1270_x_Osubalgebra__Span__incl,axiom,
! [K: set_list_a,V: set_list_a,Us: 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 @ Us ) @ V )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) @ V ) ) ) ) ).
% x.subalgebra_Span_incl
thf(fact_1271_x_OSpan__subalgebraI,axiom,
! [K: set_list_a,E: set_list_a,Us: 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 @ Us ) @ E )
=> ( ! [V5: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ V5 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ V5 )
=> ( ord_le8861187494160871172list_a @ E @ V5 ) ) )
=> ( E
= ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) ) ) ) ) ) ).
% x.Span_subalgebraI
thf(fact_1272_x_OSpan__subgroup__props_I2_J,axiom,
! [K: set_list_a,Us: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( 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 @ Us ) ) ) ) ).
% x.Span_subgroup_props(2)
thf(fact_1273_x_OSpan__subgroup__props_I3_J,axiom,
! [K: set_list_a,Us: 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 @ Us ) @ ( 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 @ Us ) )
=> ( ( member_list_a @ V22 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) )
=> ( 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 @ Us ) ) ) ) ) ) ).
% x.Span_subgroup_props(3)
thf(fact_1274_x_OSpan__smult__closed,axiom,
! [K: set_list_a,Us: list_list_a,K2: list_a,V6: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K2 @ K )
=> ( ( member_list_a @ V6 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ V6 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) ) ) ) ) ) ).
% x.Span_smult_closed
thf(fact_1275_x_OSpan__is__subalgebra,axiom,
! [K: set_list_a,Us: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( 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 @ Us ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Span_is_subalgebra
thf(fact_1276_x_Oring_Oimg__is__subfield_I2_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( subfield_a_b
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ K )
@ r ) ) ) ).
% x.ring.img_is_subfield(2)
thf(fact_1277_x_OSpan__m__inv__simprule,axiom,
! [K: set_list_a,Us: list_list_a,K2: list_a,A2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ 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 ) ) @ K2 @ A2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) )
=> ( member_list_a @ A2 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) ) ) ) ) ) ) ).
% x.Span_m_inv_simprule
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( eval_a_b @ r @ p @ x )
= ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) ) ).
%------------------------------------------------------------------------------