TPTP Problem File: SLH0158^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Interpolation_Polynomials_HOL_Algebra/0001_Lagrange_Interpolation/prob_00223_008531__17317234_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1493 ( 412 unt; 219 typ; 0 def)
% Number of atoms : 3933 (1234 equ; 0 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 15974 ( 334 ~; 77 |; 171 &;13352 @)
% ( 0 <=>;2040 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 27 ( 26 usr)
% Number of type conns : 565 ( 565 >; 0 *; 0 +; 0 <<)
% Number of symbols : 194 ( 193 usr; 12 con; 0-4 aty)
% Number of variables : 3043 ( 50 ^;2910 !; 83 ?;3043 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:38:05.870
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
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insert_set_list_a: set_list_a > set_set_list_a > set_set_list_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Subrings_Osubcring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subcri7763218559781929323t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subdom7821232466298058046t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_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_Itf__a_J,type,
bound_list_a: list_a > nat > ( nat > list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_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_001tf__a_001tf__b,type,
up_a_b: partia2175431115845679010xt_a_b > set_nat_a ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member_list_a_list_a: ( list_a > list_a ) > set_list_a_list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
member_list_a_a: ( list_a > a ) > set_list_a_a > $o ).
thf(sy_c_member_001_062_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_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
member_a_list_a: ( a > list_a ) > set_a_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
member_list_set_a: list_set_a > set_list_set_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__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 ).
% Relevant facts (1273)
thf(fact_0_factorial__domain__axioms,axiom,
ring_f5272581269873410839in_a_b @ r ).
% factorial_domain_axioms
thf(fact_1_local_Ofield__axioms,axiom,
field_a_b @ r ).
% local.field_axioms
thf(fact_2_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_3_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_4_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_5_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_6_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_7_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_8_x_Osemiring__axioms,axiom,
semiri2871908745932252451t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.semiring_axioms
thf(fact_9_poly__of__const__in__carrier,axiom,
! [S: a] :
( ( member_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( poly_of_const_a_b @ r @ S ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% poly_of_const_in_carrier
thf(fact_10_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_11_x_Oadd_Ol__cancel,axiom,
! [C: list_a,A: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A = B ) ) ) ) ) ).
% x.add.l_cancel
thf(fact_12_x_Oadd_Or__cancel,axiom,
! [A: list_a,C: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A = B ) ) ) ) ) ).
% x.add.r_cancel
thf(fact_13_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_14_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_15_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_16_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_17_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_18_x_Oadd_Oint__pow__mult__distrib,axiom,
! [X: list_a,Y: list_a,I: int] :
( ( ( 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 ) )
=> ( ( 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_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ ( 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 ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ X ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ Y ) ) ) ) ) ) ).
% x.add.int_pow_mult_distrib
thf(fact_19_x_Oadd_Oint__pow__distrib,axiom,
! [X: list_a,Y: list_a,I: int] :
( ( 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_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ ( 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 ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ X ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ Y ) ) ) ) ) ).
% x.add.int_pow_distrib
thf(fact_20_x_Ocgenideal__is__principalideal,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.cgenideal_is_principalideal
thf(fact_21_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_22_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_23_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_24_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_25_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_26_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_27_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_28_x_Oadd_Opow__mult__distrib,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( ( 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 ) )
=> ( ( 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_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ ( 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 ) ) @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ X ) @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ Y ) ) ) ) ) ) ).
% x.add.pow_mult_distrib
thf(fact_29_x_Oadd_Onat__pow__distrib,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( 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_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ ( 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 ) ) @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ X ) @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ Y ) ) ) ) ) ).
% x.add.nat_pow_distrib
thf(fact_30_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_31_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_32_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_33_x_Ocgenideal__self,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ I @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I ) ) ) ).
% x.cgenideal_self
thf(fact_34_x_Oadd_Ogroup__commutes__pow,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( ( 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 ) )
=> ( ( 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 ) ) @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ X ) @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ X ) ) ) ) ) ) ).
% x.add.group_commutes_pow
thf(fact_35_x_Oadd_Onat__pow__comm,axiom,
! [X: list_a,N: nat,M: nat] :
( ( 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 ) ) @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ X ) @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M @ X ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M @ X ) @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ X ) ) ) ) ).
% x.add.nat_pow_comm
thf(fact_36_x_Oadd__pow__ldistr,axiom,
! [A: list_a,B: list_a,K: nat] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A ) @ B )
= ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ).
% x.add_pow_ldistr
thf(fact_37_x_Oadd__pow__rdistr,axiom,
! [A: list_a,B: list_a,K: nat] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ B ) )
= ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ).
% x.add_pow_rdistr
thf(fact_38_x_Oadd__pow__ldistr__int,axiom,
! [A: list_a,B: list_a,K: int] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A ) @ B )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ).
% x.add_pow_ldistr_int
thf(fact_39_x_Oadd__pow__rdistr__int,axiom,
! [A: list_a,B: list_a,K: int] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ B ) )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ).
% x.add_pow_rdistr_int
thf(fact_40_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_41_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_42_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_mem__Collect__eq,axiom,
! [A: nat > list_a,P: ( nat > list_a ) > $o] :
( ( member_nat_list_a @ A @ ( collect_nat_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A: nat > a,P: ( nat > a ) > $o] :
( ( member_nat_a @ A @ ( collect_nat_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A2: set_nat_list_a] :
( ( collect_nat_list_a
@ ^ [X3: nat > list_a] : ( member_nat_list_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A2: set_nat_a] :
( ( collect_nat_a
@ ^ [X3: nat > a] : ( member_nat_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_x_Oadd_Onat__pow__closed,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.nat_pow_closed
thf(fact_51_x_Oadd_Onat__pow__one,axiom,
! [N: nat] :
( ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.add.nat_pow_one
thf(fact_52_x_Oadd_Oint__pow__closed,axiom,
! [X: list_a,I: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.int_pow_closed
thf(fact_53_x_Oadd_Oint__pow__one,axiom,
! [Z: int] :
( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.add.int_pow_one
thf(fact_54_x_Oadd_Ol__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one
thf(fact_55_x_Oadd_Ol__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one'
thf(fact_56_x_Oadd_Or__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one
thf(fact_57_x_Oadd_Or__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one'
thf(fact_58_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_59_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_60_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_61_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_62_x_Omonoid__cancelI,axiom,
( ! [A3: list_a,B2: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ A3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ B2 ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: list_a,B2: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ C2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B2 @ C2 ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.monoid_cancelI
thf(fact_63_is__root__poly__mult__imp__is__root,axiom,
! [P2: list_a,Q: list_a,X: a] :
( ( member_list_a @ P2 @ ( 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 ) ) @ P2 @ Q ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_64_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_65_x_Oline__extension__mem__iff,axiom,
! [U: list_a,K2: set_list_a,A: list_a,E: set_list_a] :
( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A @ E ) )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ K2 )
& ? [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 @ A ) @ Y3 ) ) ) ) ) ) ).
% x.line_extension_mem_iff
thf(fact_66_x_Oadd_Onat__pow__mult,axiom,
! [X: list_a,N: nat,M: nat] :
( ( 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 ) ) @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ X ) @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M @ X ) )
= ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( plus_plus_nat @ N @ M ) @ X ) ) ) ).
% x.add.nat_pow_mult
thf(fact_67_x_Oadd_Opow__eq__div2,axiom,
! [X: list_a,M: nat,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M @ X )
= ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ X ) )
=> ( ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( minus_minus_nat @ M @ N ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.pow_eq_div2
thf(fact_68_x_Oadd_Oint__pow__mult,axiom,
! [X: list_a,I: int,J: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( plus_plus_int @ I @ J ) @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ X ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ J @ X ) ) ) ) ).
% x.add.int_pow_mult
thf(fact_69_x_Oadd_Onat__pow__Suc2,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( suc @ N ) @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ X ) ) ) ) ).
% x.add.nat_pow_Suc2
thf(fact_70_x_Ocgenideal__prod,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( set_mu3586181839180059898t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B ) )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ).
% x.cgenideal_prod
thf(fact_71_x_Oadd_Onat__pow__pow,axiom,
! [X: list_a,M: nat,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ X ) )
= ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( times_times_nat @ N @ M ) @ X ) ) ) ).
% x.add.nat_pow_pow
thf(fact_72_x_Oadd_Oint__pow__pow,axiom,
! [X: list_a,M: int,N: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ X ) )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( times_times_int @ N @ M ) @ X ) ) ) ).
% x.add.int_pow_pow
thf(fact_73_x_Oadd_Onat__pow__Suc,axiom,
! [N: nat,X: list_a] :
( ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( suc @ N ) @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ X ) @ X ) ) ).
% x.add.nat_pow_Suc
thf(fact_74_x_Ocgenideal__eq__genideal,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I )
= ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I @ bot_bot_set_list_a ) ) ) ) ).
% x.cgenideal_eq_genideal
thf(fact_75_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_76_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_77_x_Ogenideal__self_H,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ I @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I @ bot_bot_set_list_a ) ) ) ) ).
% x.genideal_self'
thf(fact_78_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_79_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_80_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_81_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_82_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_83_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_84_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_85_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_86_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_87_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_88_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_89_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_90_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_91_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_92_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_93_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_94_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_95_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_96_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_97_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_98_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_99_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_100_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_101_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_102_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_103_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_104_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_105_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_106_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_107_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_108_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_109_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_110_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_111_x_OsubdomainI,axiom,
! [H: set_list_a] :
( ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [H1: list_a,H2: list_a] :
( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H2 )
= ( 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 ) ) ) )
| ( H2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) )
=> ( subdom7821232466298058046t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.subdomainI
thf(fact_112_x_OIdl__subset__ideal_H,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) )
= ( member_list_a @ A @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ) ) ).
% x.Idl_subset_ideal'
thf(fact_113_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_114_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
!= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_115_semiring_Ocarrier__one__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_116_semiring_Ocarrier__one__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_117_semiring_Oone__zeroI,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
=> ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_118_semiring_Oone__zeroI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_119_semiring_Oone__zeroD,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_120_semiring_Oone__zeroD,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) )
=> ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_121_x_Ocring__fieldI2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A3
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ X4 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.cring_fieldI2
thf(fact_122_principalideal_Ogenerate,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( princi8786919440553033881t_unit @ I2 @ R )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
& ( I2
= ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_123_principalideal_Ogenerate,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
& ( I2
= ( genideal_a_b @ R @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_124_ring__hom__memI,axiom,
! [R: partia2670972154091845814t_unit,H3: list_a > list_a,S2: partia2670972154091845814t_unit] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H3 @ X2 ) @ ( partia5361259788508890537t_unit @ S2 ) ) )
=> ( ! [X2: list_a,Y4: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R @ X2 @ Y4 ) )
= ( mult_l7073676228092353617t_unit @ S2 @ ( H3 @ X2 ) @ ( H3 @ Y4 ) ) ) ) )
=> ( ! [X2: list_a,Y4: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y4 ) )
= ( add_li7652885771158616974t_unit @ S2 @ ( H3 @ X2 ) @ ( H3 @ Y4 ) ) ) ) )
=> ( ( ( H3 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S2 ) )
=> ( member_list_a_list_a @ H3 @ ( ring_h7399960747407462284t_unit @ R @ S2 ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_125_ring__hom__memI,axiom,
! [R: partia2670972154091845814t_unit,H3: list_a > a,S2: partia2175431115845679010xt_a_b] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H3 @ X2 ) @ ( partia707051561876973205xt_a_b @ S2 ) ) )
=> ( ! [X2: list_a,Y4: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R @ X2 @ Y4 ) )
= ( mult_a_ring_ext_a_b @ S2 @ ( H3 @ X2 ) @ ( H3 @ Y4 ) ) ) ) )
=> ( ! [X2: list_a,Y4: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y4 ) )
= ( add_a_b @ S2 @ ( H3 @ X2 ) @ ( H3 @ Y4 ) ) ) ) )
=> ( ( ( H3 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S2 ) )
=> ( member_list_a_a @ H3 @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_126_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H3: a > list_a,S2: partia2670972154091845814t_unit] :
( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H3 @ X2 ) @ ( partia5361259788508890537t_unit @ S2 ) ) )
=> ( ! [X2: a,Y4: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y4 ) )
= ( mult_l7073676228092353617t_unit @ S2 @ ( H3 @ X2 ) @ ( H3 @ Y4 ) ) ) ) )
=> ( ! [X2: a,Y4: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( add_a_b @ R @ X2 @ Y4 ) )
= ( add_li7652885771158616974t_unit @ S2 @ ( H3 @ X2 ) @ ( H3 @ Y4 ) ) ) ) )
=> ( ( ( H3 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_li8328186300101108157t_unit @ S2 ) )
=> ( member_a_list_a @ H3 @ ( ring_h405018892823518980t_unit @ R @ S2 ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_127_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H3: a > a,S2: partia2175431115845679010xt_a_b] :
( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H3 @ X2 ) @ ( partia707051561876973205xt_a_b @ S2 ) ) )
=> ( ! [X2: a,Y4: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y4 ) )
= ( mult_a_ring_ext_a_b @ S2 @ ( H3 @ X2 ) @ ( H3 @ Y4 ) ) ) ) )
=> ( ! [X2: a,Y4: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( add_a_b @ R @ X2 @ Y4 ) )
= ( add_a_b @ S2 @ ( H3 @ X2 ) @ ( H3 @ Y4 ) ) ) ) )
=> ( ( ( H3 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S2 ) )
=> ( member_a_a @ H3 @ ( ring_hom_a_b_a_b @ R @ S2 ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_128_x_Oset__mult__closed,axiom,
! [H: set_list_a,K2: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_mu3586181839180059898t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ K2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.set_mult_closed
thf(fact_129_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_130_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_131_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_132_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_133_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_134_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_135_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_136_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_137_genideal__self_H,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_138_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_139_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_140_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_141_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_142_m__rcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B @ A )
= ( mult_a_ring_ext_a_b @ r @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% m_rcancel
thf(fact_143_m__lcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( mult_a_ring_ext_a_b @ r @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% m_lcancel
thf(fact_144_integral__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_145_local_Ointegral,axiom,
! [A: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_146_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_147_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_148_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_149_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_150_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_151_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_152_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_153_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_154_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_155_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_156_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_157_add_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( add_a_b @ r @ A @ C )
= ( add_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.r_cancel
thf(fact_158_add_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( add_a_b @ r @ C @ A )
= ( add_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.l_cancel
thf(fact_159_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_160_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_161_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A3
!= ( zero_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A3 @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_162_x_Oline__extension__in__carrier,axiom,
! [K2: set_list_a,A: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A @ E ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.line_extension_in_carrier
thf(fact_163_x_Osubset__Idl__subset,axiom,
! [I2: set_list_a,H: set_list_a] :
( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ H @ I2 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 ) ) ) ) ).
% x.subset_Idl_subset
thf(fact_164_x_Ogenideal__self,axiom,
! [S2: set_list_a] :
( ( ord_le8861187494160871172list_a @ S2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ S2 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) ) ) ).
% x.genideal_self
thf(fact_165_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_166_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_167_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_168_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_169_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_170_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_171_add_Or__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ A @ X ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_172_add_Or__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_173_add_Ol__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ X @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_174_add_Ol__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ A )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_175_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_176_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_177_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_178_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_179_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_180_lift__Suc__antimono__le,axiom,
! [F: nat > set_list_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le8861187494160871172list_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le8861187494160871172list_a @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_181_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_182_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_183_lift__Suc__mono__le,axiom,
! [F: nat > set_list_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le8861187494160871172list_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le8861187494160871172list_a @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_184_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_set_a @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_185_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_186_principalideal_Ois__principalideal,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( princi8786919440553033881t_unit @ I2 @ R )
=> ( princi8786919440553033881t_unit @ I2 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_187_principalideal_Ois__principalideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ( principalideal_a_b @ I2 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_188_ring__hom__closed,axiom,
! [H3: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H3 @ X ) @ ( partia5361259788508890537t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_189_ring__hom__closed,axiom,
! [H3: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a] :
( ( member_list_a_a @ H3 @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H3 @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_190_ring__hom__closed,axiom,
! [H3: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a] :
( ( member_a_list_a @ H3 @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H3 @ X ) @ ( partia5361259788508890537t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_191_ring__hom__closed,axiom,
! [H3: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H3 @ ( ring_hom_a_b_a_b @ R @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H3 @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_192_ring__hom__one,axiom,
! [H3: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit] :
( ( member_list_a_list_a @ H3 @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
=> ( ( H3 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S2 ) ) ) ).
% ring_hom_one
thf(fact_193_ring__hom__one,axiom,
! [H3: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b] :
( ( member_list_a_a @ H3 @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
=> ( ( H3 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S2 ) ) ) ).
% ring_hom_one
thf(fact_194_ring__hom__one,axiom,
! [H3: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H3 @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
=> ( ( H3 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_li8328186300101108157t_unit @ S2 ) ) ) ).
% ring_hom_one
thf(fact_195_ring__hom__one,axiom,
! [H3: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H3 @ ( ring_hom_a_b_a_b @ R @ S2 ) )
=> ( ( H3 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S2 ) ) ) ).
% ring_hom_one
thf(fact_196_ring__hom__add,axiom,
! [H3: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S2 @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_197_ring__hom__add,axiom,
! [H3: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H3 @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_a_b @ S2 @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_198_ring__hom__add,axiom,
! [H3: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H3 @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( add_a_b @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S2 @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_199_ring__hom__add,axiom,
! [H3: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H3 @ ( ring_hom_a_b_a_b @ R @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S2 @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_200_ring__hom__mult,axiom,
! [H3: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S2 @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_201_ring__hom__mult,axiom,
! [H3: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H3 @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S2 @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_202_ring__hom__mult,axiom,
! [H3: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H3 @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S2 @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_203_ring__hom__mult,axiom,
! [H3: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H3 @ ( ring_hom_a_b_a_b @ R @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S2 @ ( H3 @ X ) @ ( H3 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_204_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_205_Ring_Oone__not__zero,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_206_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_207_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_208_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_209_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_210_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_211_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_212_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_213_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_214_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_215_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_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_Ring_Ointegral,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ A @ B )
= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( A
= ( zero_s2174465271003423091t_unit @ R ) )
| ( B
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_225_Ring_Ointegral,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( A
= ( zero_s2910681146719230829t_unit @ R ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_226_Ring_Ointegral,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( A
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_227_Ring_Ointegral,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( field_a_b @ R )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_228_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_229_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_230_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_231_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_232_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_233_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_234_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_235_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_236_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_237_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_238_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_239_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_240_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_241_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_242_semiring_Oadd__pow__ldistr,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,K: nat] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( add_po2640537187477214989it_nat @ R @ K @ A ) @ B )
= ( add_po2640537187477214989it_nat @ R @ K @ ( mult_l7073676228092353617t_unit @ R @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_ldistr
thf(fact_243_semiring_Oadd__pow__ldistr,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,K: nat] :
( ( semiring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_pow_a_b_nat @ R @ K @ A ) @ B )
= ( add_pow_a_b_nat @ R @ K @ ( mult_a_ring_ext_a_b @ R @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_ldistr
thf(fact_244_semiring_Oadd__pow__rdistr,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,K: nat] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ A @ ( add_po2640537187477214989it_nat @ R @ K @ B ) )
= ( add_po2640537187477214989it_nat @ R @ K @ ( mult_l7073676228092353617t_unit @ R @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_rdistr
thf(fact_245_semiring_Oadd__pow__rdistr,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,K: nat] :
( ( semiring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ A @ ( add_pow_a_b_nat @ R @ K @ B ) )
= ( add_pow_a_b_nat @ R @ K @ ( mult_a_ring_ext_a_b @ R @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_rdistr
thf(fact_246_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_247_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_248_singleton__insert__inj__eq_H,axiom,
! [A: list_a,A2: set_list_a,B: list_a] :
( ( ( insert_list_a @ A @ A2 )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_249_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_250_singleton__insert__inj__eq,axiom,
! [B: list_a,A: list_a,A2: set_list_a] :
( ( ( insert_list_a @ B @ bot_bot_set_list_a )
= ( insert_list_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_251_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_252_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_253_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_254_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_255_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_256_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_257_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_258_x_Oa__lcos__m__assoc,axiom,
! [M2: set_list_a,G: list_a,H3: list_a] :
( ( ord_le8861187494160871172list_a @ M2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ H3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H3 @ M2 ) )
= ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ H3 ) @ M2 ) ) ) ) ) ).
% x.a_lcos_m_assoc
thf(fact_259_assms_I2_J,axiom,
ord_less_eq_set_a @ s @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(2)
thf(fact_260_add_Ogroup__commutes__pow,axiom,
! [X: a,Y: a,N: nat] :
( ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ Y )
= ( add_a_b @ r @ Y @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ) ) ).
% add.group_commutes_pow
thf(fact_261_add_Onat__pow__comm,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ M @ X ) )
= ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ M @ X ) @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ).
% add.nat_pow_comm
thf(fact_262_add_Onat__pow__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ N @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ N @ Y ) ) ) ) ) ).
% add.nat_pow_distrib
thf(fact_263_add_Opow__mult__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ N @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ N @ Y ) ) ) ) ) ) ).
% add.pow_mult_distrib
thf(fact_264_add_Onat__pow__pow,axiom,
! [X: a,M: nat,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ M @ ( add_pow_a_b_nat @ r @ N @ X ) )
= ( add_pow_a_b_nat @ r @ ( times_times_nat @ N @ M ) @ X ) ) ) ).
% add.nat_pow_pow
thf(fact_265_add__pow__ldistr,axiom,
! [A: a,B: a,K: nat] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_nat @ r @ K @ A ) @ B )
= ( add_pow_a_b_nat @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_ldistr
thf(fact_266_add__pow__rdistr,axiom,
! [A: a,B: a,K: nat] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_nat @ r @ K @ B ) )
= ( add_pow_a_b_nat @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% add_pow_rdistr
thf(fact_267_genideal__self,axiom,
! [S2: set_a] :
( ( ord_less_eq_set_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ S2 @ ( genideal_a_b @ r @ S2 ) ) ) ).
% genideal_self
thf(fact_268_subset__Idl__subset,axiom,
! [I2: set_a,H: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H ) @ ( genideal_a_b @ r @ I2 ) ) ) ) ).
% subset_Idl_subset
thf(fact_269_add_Onat__pow__Suc2,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_pow_a_b_nat @ r @ ( suc @ N ) @ X )
= ( add_a_b @ r @ X @ ( add_pow_a_b_nat @ r @ N @ X ) ) ) ) ).
% add.nat_pow_Suc2
thf(fact_270_add_Opow__eq__div2,axiom,
! [X: a,M: nat,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_pow_a_b_nat @ r @ M @ X )
= ( add_pow_a_b_nat @ r @ N @ X ) )
=> ( ( add_pow_a_b_nat @ r @ ( minus_minus_nat @ M @ N ) @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.pow_eq_div2
thf(fact_271_add_Onat__pow__mult,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( add_pow_a_b_nat @ r @ M @ X ) )
= ( add_pow_a_b_nat @ r @ ( plus_plus_nat @ N @ M ) @ X ) ) ) ).
% add.nat_pow_mult
thf(fact_272_subset__antisym,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_273_subset__antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_274_subsetI,axiom,
! [A2: set_nat_list_a,B3: set_nat_list_a] :
( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A2 )
=> ( member_nat_list_a @ X2 @ B3 ) )
=> ( ord_le2145805922479659755list_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_275_subsetI,axiom,
! [A2: set_nat_a,B3: set_nat_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_nat_a @ X2 @ B3 ) )
=> ( ord_le871467723717165285_nat_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_276_subsetI,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( member_list_a @ X2 @ B3 ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_277_subsetI,axiom,
! [A2: set_a,B3: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B3 ) )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_278_Diff__cancel,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ A2 )
= bot_bot_set_list_a ) ).
% Diff_cancel
thf(fact_279_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_280_empty__Diff,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ bot_bot_set_list_a @ A2 )
= bot_bot_set_list_a ) ).
% empty_Diff
thf(fact_281_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_282_Diff__empty,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% Diff_empty
thf(fact_283_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_284_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_285_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_286_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_287_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_288_all__not__in__conv,axiom,
! [A2: set_nat_list_a] :
( ( ! [X3: nat > list_a] :
~ ( member_nat_list_a @ X3 @ A2 ) )
= ( A2 = bot_bo3806784159821827511list_a ) ) ).
% all_not_in_conv
thf(fact_289_all__not__in__conv,axiom,
! [A2: set_nat_a] :
( ( ! [X3: nat > a] :
~ ( member_nat_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat_a ) ) ).
% all_not_in_conv
thf(fact_290_all__not__in__conv,axiom,
! [A2: set_list_a] :
( ( ! [X3: list_a] :
~ ( member_list_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_291_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_292_empty__iff,axiom,
! [C: nat > list_a] :
~ ( member_nat_list_a @ C @ bot_bo3806784159821827511list_a ) ).
% empty_iff
thf(fact_293_empty__iff,axiom,
! [C: nat > a] :
~ ( member_nat_a @ C @ bot_bot_set_nat_a ) ).
% empty_iff
thf(fact_294_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_295_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_296_insert__Diff1,axiom,
! [X: nat > list_a,B3: set_nat_list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ X @ B3 )
=> ( ( minus_4169782841487898290list_a @ ( insert_nat_list_a @ X @ A2 ) @ B3 )
= ( minus_4169782841487898290list_a @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_297_insert__Diff1,axiom,
! [X: nat > a,B3: set_nat_a,A2: set_nat_a] :
( ( member_nat_a @ X @ B3 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ B3 )
= ( minus_490503922182417452_nat_a @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_298_insert__Diff1,axiom,
! [X: list_a,B3: set_list_a,A2: set_list_a] :
( ( member_list_a @ X @ B3 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A2 ) @ B3 )
= ( minus_646659088055828811list_a @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_299_insert__Diff1,axiom,
! [X: a,B3: set_a,A2: set_a] :
( ( member_a @ X @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_300_Diff__insert0,axiom,
! [X: nat > list_a,A2: set_nat_list_a,B3: set_nat_list_a] :
( ~ ( member_nat_list_a @ X @ A2 )
=> ( ( minus_4169782841487898290list_a @ A2 @ ( insert_nat_list_a @ X @ B3 ) )
= ( minus_4169782841487898290list_a @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_301_Diff__insert0,axiom,
! [X: nat > a,A2: set_nat_a,B3: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ X @ B3 ) )
= ( minus_490503922182417452_nat_a @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_302_Diff__insert0,axiom,
! [X: list_a,A2: set_list_a,B3: set_list_a] :
( ~ ( member_list_a @ X @ A2 )
=> ( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ B3 ) )
= ( minus_646659088055828811list_a @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_303_Diff__insert0,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_304_insert__absorb2,axiom,
! [X: list_a,A2: set_list_a] :
( ( insert_list_a @ X @ ( insert_list_a @ X @ A2 ) )
= ( insert_list_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_305_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_306_insert__iff,axiom,
! [A: list_a,B: list_a,A2: set_list_a] :
( ( member_list_a @ A @ ( insert_list_a @ B @ A2 ) )
= ( ( A = B )
| ( member_list_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_307_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_308_insert__iff,axiom,
! [A: nat > list_a,B: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ A @ ( insert_nat_list_a @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_list_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_309_insert__iff,axiom,
! [A: nat > a,B: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ ( insert_nat_a @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_310_insertCI,axiom,
! [A: list_a,B3: set_list_a,B: list_a] :
( ( ~ ( member_list_a @ A @ B3 )
=> ( A = B ) )
=> ( member_list_a @ A @ ( insert_list_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_311_insertCI,axiom,
! [A: a,B3: set_a,B: a] :
( ( ~ ( member_a @ A @ B3 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_312_insertCI,axiom,
! [A: nat > list_a,B3: set_nat_list_a,B: nat > list_a] :
( ( ~ ( member_nat_list_a @ A @ B3 )
=> ( A = B ) )
=> ( member_nat_list_a @ A @ ( insert_nat_list_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_313_insertCI,axiom,
! [A: nat > a,B3: set_nat_a,B: nat > a] :
( ( ~ ( member_nat_a @ A @ B3 )
=> ( A = B ) )
=> ( member_nat_a @ A @ ( insert_nat_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_314_Idl__subset__ideal_H,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_315_zeromaximalideal,axiom,
maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeromaximalideal
thf(fact_316_x_Oa__l__coset__subset__G,axiom,
! [H: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( 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 @ H ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.a_l_coset_subset_G
thf(fact_317_x_Oa__lcos__mult__one,axiom,
! [M2: set_list_a] :
( ( ord_le8861187494160871172list_a @ M2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ M2 )
= M2 ) ) ).
% x.a_lcos_mult_one
thf(fact_318_Diff__eq__empty__iff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ( minus_646659088055828811list_a @ A2 @ B3 )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_319_Diff__eq__empty__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ( minus_minus_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_320_empty__subsetI,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% empty_subsetI
thf(fact_321_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_322_subset__empty,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_323_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_324_insert__subset,axiom,
! [X: nat > list_a,A2: set_nat_list_a,B3: set_nat_list_a] :
( ( ord_le2145805922479659755list_a @ ( insert_nat_list_a @ X @ A2 ) @ B3 )
= ( ( member_nat_list_a @ X @ B3 )
& ( ord_le2145805922479659755list_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_325_insert__subset,axiom,
! [X: nat > a,A2: set_nat_a,B3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( insert_nat_a @ X @ A2 ) @ B3 )
= ( ( member_nat_a @ X @ B3 )
& ( ord_le871467723717165285_nat_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_326_insert__subset,axiom,
! [X: list_a,A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X @ A2 ) @ B3 )
= ( ( member_list_a @ X @ B3 )
& ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_327_insert__subset,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( ( member_a @ X @ B3 )
& ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_328_insert__Diff__single,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
= ( insert_list_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_329_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_330_singletonI,axiom,
! [A: nat > list_a] : ( member_nat_list_a @ A @ ( insert_nat_list_a @ A @ bot_bo3806784159821827511list_a ) ) ).
% singletonI
thf(fact_331_singletonI,axiom,
! [A: nat > a] : ( member_nat_a @ A @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) ) ).
% singletonI
thf(fact_332_singletonI,axiom,
! [A: list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_333_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_334_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_335_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_336_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_337_add_Onat__pow__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_pow_a_b_nat @ r @ N @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% add.nat_pow_closed
thf(fact_338_add_Onat__pow__one,axiom,
! [N: nat] :
( ( add_pow_a_b_nat @ r @ N @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% add.nat_pow_one
thf(fact_339_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_340_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_341_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_342_add_Onat__pow__Suc,axiom,
! [N: nat,X: a] :
( ( add_pow_a_b_nat @ r @ ( suc @ N ) @ X )
= ( add_a_b @ r @ ( add_pow_a_b_nat @ r @ N @ X ) @ X ) ) ).
% add.nat_pow_Suc
thf(fact_343_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_344_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_345_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_346_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_347_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_348_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_349_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_350_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_351_maximalideal_Ois__maximalideal,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( maxima6585700282301356660t_unit @ I2 @ R )
=> ( maxima6585700282301356660t_unit @ I2 @ R ) ) ).
% maximalideal.is_maximalideal
thf(fact_352_maximalideal_Ois__maximalideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( maximalideal_a_b @ I2 @ R )
=> ( maximalideal_a_b @ I2 @ R ) ) ).
% maximalideal.is_maximalideal
thf(fact_353_double__diff,axiom,
! [A2: set_list_a,B3: set_list_a,C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C3 )
=> ( ( minus_646659088055828811list_a @ B3 @ ( minus_646659088055828811list_a @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_354_double__diff,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C3 )
=> ( ( minus_minus_set_a @ B3 @ ( minus_minus_set_a @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_355_Diff__subset,axiom,
! [A2: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_356_Diff__subset,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_357_Diff__mono,axiom,
! [A2: set_list_a,C3: set_list_a,D: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ C3 )
=> ( ( ord_le8861187494160871172list_a @ D @ B3 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) @ ( minus_646659088055828811list_a @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_358_Diff__mono,axiom,
! [A2: set_a,C3: set_a,D: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ D @ B3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( minus_minus_set_a @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_359_insert__Diff__if,axiom,
! [X: nat > list_a,B3: set_nat_list_a,A2: set_nat_list_a] :
( ( ( member_nat_list_a @ X @ B3 )
=> ( ( minus_4169782841487898290list_a @ ( insert_nat_list_a @ X @ A2 ) @ B3 )
= ( minus_4169782841487898290list_a @ A2 @ B3 ) ) )
& ( ~ ( member_nat_list_a @ X @ B3 )
=> ( ( minus_4169782841487898290list_a @ ( insert_nat_list_a @ X @ A2 ) @ B3 )
= ( insert_nat_list_a @ X @ ( minus_4169782841487898290list_a @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_360_insert__Diff__if,axiom,
! [X: nat > a,B3: set_nat_a,A2: set_nat_a] :
( ( ( member_nat_a @ X @ B3 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ B3 )
= ( minus_490503922182417452_nat_a @ A2 @ B3 ) ) )
& ( ~ ( member_nat_a @ X @ B3 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ B3 )
= ( insert_nat_a @ X @ ( minus_490503922182417452_nat_a @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_361_insert__Diff__if,axiom,
! [X: list_a,B3: set_list_a,A2: set_list_a] :
( ( ( member_list_a @ X @ B3 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A2 ) @ B3 )
= ( minus_646659088055828811list_a @ A2 @ B3 ) ) )
& ( ~ ( member_list_a @ X @ B3 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A2 ) @ B3 )
= ( insert_list_a @ X @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_362_insert__Diff__if,axiom,
! [X: a,B3: set_a,A2: set_a] :
( ( ( member_a @ X @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( minus_minus_set_a @ A2 @ B3 ) ) )
& ( ~ ( member_a @ X @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_363_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y4: nat,Z2: nat] :
( ( R @ X2 @ Y4 )
=> ( ( R @ Y4 @ Z2 )
=> ( R @ X2 @ Z2 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_364_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_365_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_366_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_367_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_368_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_369_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_370_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_371_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_372_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_373_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_374_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_375_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_376_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_377_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_378_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_379_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_380_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_381_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_382_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_383_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_384_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_385_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_386_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_387_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_388_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_389_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_390_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_391_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_392_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_393_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_394_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_395_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_396_subset__Diff__insert,axiom,
! [A2: set_nat_list_a,B3: set_nat_list_a,X: nat > list_a,C3: set_nat_list_a] :
( ( ord_le2145805922479659755list_a @ A2 @ ( minus_4169782841487898290list_a @ B3 @ ( insert_nat_list_a @ X @ C3 ) ) )
= ( ( ord_le2145805922479659755list_a @ A2 @ ( minus_4169782841487898290list_a @ B3 @ C3 ) )
& ~ ( member_nat_list_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_397_subset__Diff__insert,axiom,
! [A2: set_nat_a,B3: set_nat_a,X: nat > a,C3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A2 @ ( minus_490503922182417452_nat_a @ B3 @ ( insert_nat_a @ X @ C3 ) ) )
= ( ( ord_le871467723717165285_nat_a @ A2 @ ( minus_490503922182417452_nat_a @ B3 @ C3 ) )
& ~ ( member_nat_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_398_subset__Diff__insert,axiom,
! [A2: set_list_a,B3: set_list_a,X: list_a,C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( minus_646659088055828811list_a @ B3 @ ( insert_list_a @ X @ C3 ) ) )
= ( ( ord_le8861187494160871172list_a @ A2 @ ( minus_646659088055828811list_a @ B3 @ C3 ) )
& ~ ( member_list_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_399_subset__Diff__insert,axiom,
! [A2: set_a,B3: set_a,X: a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B3 @ ( insert_a @ X @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B3 @ C3 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_400_Diff__insert,axiom,
! [A2: set_list_a,A: list_a,B3: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B3 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ).
% Diff_insert
thf(fact_401_Diff__insert,axiom,
! [A2: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_402_insert__Diff,axiom,
! [A: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ A @ A2 )
=> ( ( insert_nat_list_a @ A @ ( minus_4169782841487898290list_a @ A2 @ ( insert_nat_list_a @ A @ bot_bo3806784159821827511list_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_403_insert__Diff,axiom,
! [A: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ A2 )
=> ( ( insert_nat_a @ A @ ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_404_insert__Diff,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_405_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_406_Diff__insert2,axiom,
! [A2: set_list_a,A: list_a,B3: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B3 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_407_Diff__insert2,axiom,
! [A2: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_408_Diff__insert__absorb,axiom,
! [X: nat > list_a,A2: set_nat_list_a] :
( ~ ( member_nat_list_a @ X @ A2 )
=> ( ( minus_4169782841487898290list_a @ ( insert_nat_list_a @ X @ A2 ) @ ( insert_nat_list_a @ X @ bot_bo3806784159821827511list_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_409_Diff__insert__absorb,axiom,
! [X: nat > a,A2: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ ( insert_nat_a @ X @ bot_bot_set_nat_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_410_Diff__insert__absorb,axiom,
! [X: list_a,A2: set_list_a] :
( ~ ( member_list_a @ X @ A2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A2 ) @ ( insert_list_a @ X @ bot_bot_set_list_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_411_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_412_maximalideal_OI__notcarr,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( maxima6585700282301356660t_unit @ I2 @ R )
=> ( ( partia5361259788508890537t_unit @ R )
!= I2 ) ) ).
% maximalideal.I_notcarr
thf(fact_413_maximalideal_OI__notcarr,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( maximalideal_a_b @ I2 @ R )
=> ( ( partia707051561876973205xt_a_b @ R )
!= I2 ) ) ).
% maximalideal.I_notcarr
thf(fact_414_subset__insert__iff,axiom,
! [A2: set_nat_list_a,X: nat > list_a,B3: set_nat_list_a] :
( ( ord_le2145805922479659755list_a @ A2 @ ( insert_nat_list_a @ X @ B3 ) )
= ( ( ( member_nat_list_a @ X @ A2 )
=> ( ord_le2145805922479659755list_a @ ( minus_4169782841487898290list_a @ A2 @ ( insert_nat_list_a @ X @ bot_bo3806784159821827511list_a ) ) @ B3 ) )
& ( ~ ( member_nat_list_a @ X @ A2 )
=> ( ord_le2145805922479659755list_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_415_subset__insert__iff,axiom,
! [A2: set_nat_a,X: nat > a,B3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A2 @ ( insert_nat_a @ X @ B3 ) )
= ( ( ( member_nat_a @ X @ A2 )
=> ( ord_le871467723717165285_nat_a @ ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ X @ bot_bot_set_nat_a ) ) @ B3 ) )
& ( ~ ( member_nat_a @ X @ A2 )
=> ( ord_le871467723717165285_nat_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_416_subset__insert__iff,axiom,
! [A2: set_list_a,X: list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ B3 ) )
= ( ( ( member_list_a @ X @ A2 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B3 ) )
& ( ~ ( member_list_a @ X @ A2 )
=> ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_417_subset__insert__iff,axiom,
! [A2: set_a,X: a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_418_Diff__single__insert,axiom,
! [A2: set_list_a,X: list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B3 )
=> ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_419_Diff__single__insert,axiom,
! [A2: set_a,X: a,B3: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_420_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_421_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_422_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_423_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_424_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_425_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_426_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_427_Collect__mono__iff,axiom,
! [P: list_a > $o,Q2: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q2 ) )
= ( ! [X3: list_a] :
( ( P @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_428_Collect__mono__iff,axiom,
! [P: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q2 ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q2 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_429_set__eq__subset,axiom,
( ( ^ [Y6: set_list_a,Z3: set_list_a] : ( Y6 = Z3 ) )
= ( ^ [A4: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B4 )
& ( ord_le8861187494160871172list_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_430_set__eq__subset,axiom,
( ( ^ [Y6: set_a,Z3: set_a] : ( Y6 = Z3 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_431_subset__trans,axiom,
! [A2: set_list_a,B3: set_list_a,C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C3 )
=> ( ord_le8861187494160871172list_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_432_subset__trans,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C3 )
=> ( ord_less_eq_set_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_433_Collect__mono,axiom,
! [P: list_a > $o,Q2: list_a > $o] :
( ! [X2: list_a] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_434_Collect__mono,axiom,
! [P: a > $o,Q2: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q2 @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_435_subset__refl,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_436_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_437_subset__iff,axiom,
( ord_le2145805922479659755list_a
= ( ^ [A4: set_nat_list_a,B4: set_nat_list_a] :
! [T: nat > list_a] :
( ( member_nat_list_a @ T @ A4 )
=> ( member_nat_list_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_438_subset__iff,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A4: set_nat_a,B4: set_nat_a] :
! [T: nat > a] :
( ( member_nat_a @ T @ A4 )
=> ( member_nat_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_439_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B4: set_list_a] :
! [T: list_a] :
( ( member_list_a @ T @ A4 )
=> ( member_list_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_440_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T: a] :
( ( member_a @ T @ A4 )
=> ( member_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_441_equalityD2,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( A2 = B3 )
=> ( ord_le8861187494160871172list_a @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_442_equalityD2,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_443_equalityD1,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( A2 = B3 )
=> ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_444_equalityD1,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_445_subset__eq,axiom,
( ord_le2145805922479659755list_a
= ( ^ [A4: set_nat_list_a,B4: set_nat_list_a] :
! [X3: nat > list_a] :
( ( member_nat_list_a @ X3 @ A4 )
=> ( member_nat_list_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_446_subset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A4: set_nat_a,B4: set_nat_a] :
! [X3: nat > a] :
( ( member_nat_a @ X3 @ A4 )
=> ( member_nat_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_447_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B4: set_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ A4 )
=> ( member_list_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_448_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A4 )
=> ( member_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_449_equalityE,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( A2 = B3 )
=> ~ ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ~ ( ord_le8861187494160871172list_a @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_450_equalityE,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_451_subsetD,axiom,
! [A2: set_nat_list_a,B3: set_nat_list_a,C: nat > list_a] :
( ( ord_le2145805922479659755list_a @ A2 @ B3 )
=> ( ( member_nat_list_a @ C @ A2 )
=> ( member_nat_list_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_452_subsetD,axiom,
! [A2: set_nat_a,B3: set_nat_a,C: nat > a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B3 )
=> ( ( member_nat_a @ C @ A2 )
=> ( member_nat_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_453_subsetD,axiom,
! [A2: set_list_a,B3: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_454_subsetD,axiom,
! [A2: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_455_in__mono,axiom,
! [A2: set_nat_list_a,B3: set_nat_list_a,X: nat > list_a] :
( ( ord_le2145805922479659755list_a @ A2 @ B3 )
=> ( ( member_nat_list_a @ X @ A2 )
=> ( member_nat_list_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_456_in__mono,axiom,
! [A2: set_nat_a,B3: set_nat_a,X: nat > a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B3 )
=> ( ( member_nat_a @ X @ A2 )
=> ( member_nat_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_457_in__mono,axiom,
! [A2: set_list_a,B3: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( member_list_a @ X @ A2 )
=> ( member_list_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_458_in__mono,axiom,
! [A2: set_a,B3: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_459_ex__in__conv,axiom,
! [A2: set_nat_list_a] :
( ( ? [X3: nat > list_a] : ( member_nat_list_a @ X3 @ A2 ) )
= ( A2 != bot_bo3806784159821827511list_a ) ) ).
% ex_in_conv
thf(fact_460_ex__in__conv,axiom,
! [A2: set_nat_a] :
( ( ? [X3: nat > a] : ( member_nat_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat_a ) ) ).
% ex_in_conv
thf(fact_461_ex__in__conv,axiom,
! [A2: set_list_a] :
( ( ? [X3: list_a] : ( member_list_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_462_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_463_equals0I,axiom,
! [A2: set_nat_list_a] :
( ! [Y4: nat > list_a] :
~ ( member_nat_list_a @ Y4 @ A2 )
=> ( A2 = bot_bo3806784159821827511list_a ) ) ).
% equals0I
thf(fact_464_equals0I,axiom,
! [A2: set_nat_a] :
( ! [Y4: nat > a] :
~ ( member_nat_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat_a ) ) ).
% equals0I
thf(fact_465_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y4: list_a] :
~ ( member_list_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_466_equals0I,axiom,
! [A2: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_467_equals0D,axiom,
! [A2: set_nat_list_a,A: nat > list_a] :
( ( A2 = bot_bo3806784159821827511list_a )
=> ~ ( member_nat_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_468_equals0D,axiom,
! [A2: set_nat_a,A: nat > a] :
( ( A2 = bot_bot_set_nat_a )
=> ~ ( member_nat_a @ A @ A2 ) ) ).
% equals0D
thf(fact_469_equals0D,axiom,
! [A2: set_list_a,A: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_470_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_471_emptyE,axiom,
! [A: nat > list_a] :
~ ( member_nat_list_a @ A @ bot_bo3806784159821827511list_a ) ).
% emptyE
thf(fact_472_emptyE,axiom,
! [A: nat > a] :
~ ( member_nat_a @ A @ bot_bot_set_nat_a ) ).
% emptyE
thf(fact_473_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_474_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_475_mk__disjoint__insert,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ? [B5: set_list_a] :
( ( A2
= ( insert_list_a @ A @ B5 ) )
& ~ ( member_list_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_476_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B5: set_a] :
( ( A2
= ( insert_a @ A @ B5 ) )
& ~ ( member_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_477_mk__disjoint__insert,axiom,
! [A: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ A @ A2 )
=> ? [B5: set_nat_list_a] :
( ( A2
= ( insert_nat_list_a @ A @ B5 ) )
& ~ ( member_nat_list_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_478_mk__disjoint__insert,axiom,
! [A: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ A2 )
=> ? [B5: set_nat_a] :
( ( A2
= ( insert_nat_a @ A @ B5 ) )
& ~ ( member_nat_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_479_insert__commute,axiom,
! [X: list_a,Y: list_a,A2: set_list_a] :
( ( insert_list_a @ X @ ( insert_list_a @ Y @ A2 ) )
= ( insert_list_a @ Y @ ( insert_list_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_480_insert__commute,axiom,
! [X: a,Y: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A2 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_481_insert__eq__iff,axiom,
! [A: list_a,A2: set_list_a,B: list_a,B3: set_list_a] :
( ~ ( member_list_a @ A @ A2 )
=> ( ~ ( member_list_a @ B @ B3 )
=> ( ( ( insert_list_a @ A @ A2 )
= ( insert_list_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_list_a] :
( ( A2
= ( insert_list_a @ B @ C4 ) )
& ~ ( member_list_a @ B @ C4 )
& ( B3
= ( insert_list_a @ A @ C4 ) )
& ~ ( member_list_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_482_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B3: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B3 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_a] :
( ( A2
= ( insert_a @ B @ C4 ) )
& ~ ( member_a @ B @ C4 )
& ( B3
= ( insert_a @ A @ C4 ) )
& ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_483_insert__eq__iff,axiom,
! [A: nat > list_a,A2: set_nat_list_a,B: nat > list_a,B3: set_nat_list_a] :
( ~ ( member_nat_list_a @ A @ A2 )
=> ( ~ ( member_nat_list_a @ B @ B3 )
=> ( ( ( insert_nat_list_a @ A @ A2 )
= ( insert_nat_list_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_nat_list_a] :
( ( A2
= ( insert_nat_list_a @ B @ C4 ) )
& ~ ( member_nat_list_a @ B @ C4 )
& ( B3
= ( insert_nat_list_a @ A @ C4 ) )
& ~ ( member_nat_list_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_484_insert__eq__iff,axiom,
! [A: nat > a,A2: set_nat_a,B: nat > a,B3: set_nat_a] :
( ~ ( member_nat_a @ A @ A2 )
=> ( ~ ( member_nat_a @ B @ B3 )
=> ( ( ( insert_nat_a @ A @ A2 )
= ( insert_nat_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_nat_a] :
( ( A2
= ( insert_nat_a @ B @ C4 ) )
& ~ ( member_nat_a @ B @ C4 )
& ( B3
= ( insert_nat_a @ A @ C4 ) )
& ~ ( member_nat_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_485_insert__absorb,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( insert_list_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_486_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_487_insert__absorb,axiom,
! [A: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ A @ A2 )
=> ( ( insert_nat_list_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_488_insert__absorb,axiom,
! [A: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ A2 )
=> ( ( insert_nat_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_489_insert__ident,axiom,
! [X: list_a,A2: set_list_a,B3: set_list_a] :
( ~ ( member_list_a @ X @ A2 )
=> ( ~ ( member_list_a @ X @ B3 )
=> ( ( ( insert_list_a @ X @ A2 )
= ( insert_list_a @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_490_insert__ident,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B3 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_491_insert__ident,axiom,
! [X: nat > list_a,A2: set_nat_list_a,B3: set_nat_list_a] :
( ~ ( member_nat_list_a @ X @ A2 )
=> ( ~ ( member_nat_list_a @ X @ B3 )
=> ( ( ( insert_nat_list_a @ X @ A2 )
= ( insert_nat_list_a @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_492_insert__ident,axiom,
! [X: nat > a,A2: set_nat_a,B3: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ~ ( member_nat_a @ X @ B3 )
=> ( ( ( insert_nat_a @ X @ A2 )
= ( insert_nat_a @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_493_Set_Oset__insert,axiom,
! [X: list_a,A2: set_list_a] :
( ( member_list_a @ X @ A2 )
=> ~ ! [B5: set_list_a] :
( ( A2
= ( insert_list_a @ X @ B5 ) )
=> ( member_list_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_494_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B5: set_a] :
( ( A2
= ( insert_a @ X @ B5 ) )
=> ( member_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_495_Set_Oset__insert,axiom,
! [X: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ X @ A2 )
=> ~ ! [B5: set_nat_list_a] :
( ( A2
= ( insert_nat_list_a @ X @ B5 ) )
=> ( member_nat_list_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_496_Set_Oset__insert,axiom,
! [X: nat > a,A2: set_nat_a] :
( ( member_nat_a @ X @ A2 )
=> ~ ! [B5: set_nat_a] :
( ( A2
= ( insert_nat_a @ X @ B5 ) )
=> ( member_nat_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_497_insertI2,axiom,
! [A: list_a,B3: set_list_a,B: list_a] :
( ( member_list_a @ A @ B3 )
=> ( member_list_a @ A @ ( insert_list_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_498_insertI2,axiom,
! [A: a,B3: set_a,B: a] :
( ( member_a @ A @ B3 )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_499_insertI2,axiom,
! [A: nat > list_a,B3: set_nat_list_a,B: nat > list_a] :
( ( member_nat_list_a @ A @ B3 )
=> ( member_nat_list_a @ A @ ( insert_nat_list_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_500_insertI2,axiom,
! [A: nat > a,B3: set_nat_a,B: nat > a] :
( ( member_nat_a @ A @ B3 )
=> ( member_nat_a @ A @ ( insert_nat_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_501_insertI1,axiom,
! [A: list_a,B3: set_list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ B3 ) ) ).
% insertI1
thf(fact_502_insertI1,axiom,
! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% insertI1
thf(fact_503_insertI1,axiom,
! [A: nat > list_a,B3: set_nat_list_a] : ( member_nat_list_a @ A @ ( insert_nat_list_a @ A @ B3 ) ) ).
% insertI1
thf(fact_504_insertI1,axiom,
! [A: nat > a,B3: set_nat_a] : ( member_nat_a @ A @ ( insert_nat_a @ A @ B3 ) ) ).
% insertI1
thf(fact_505_insertE,axiom,
! [A: list_a,B: list_a,A2: set_list_a] :
( ( member_list_a @ A @ ( insert_list_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_list_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_506_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_507_insertE,axiom,
! [A: nat > list_a,B: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ A @ ( insert_nat_list_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat_list_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_508_insertE,axiom,
! [A: nat > a,B: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ ( insert_nat_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_509_field_Ozeromaximalideal,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( maxima2253313296322093082t_unit @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) @ R ) ) ).
% field.zeromaximalideal
thf(fact_510_field_Ozeromaximalideal,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( maxima3875439991530298004t_unit @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) @ R ) ) ).
% field.zeromaximalideal
thf(fact_511_field_Ozeromaximalideal,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R ) ) ).
% field.zeromaximalideal
thf(fact_512_field_Ozeromaximalideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ).
% field.zeromaximalideal
thf(fact_513_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_514_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_515_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_516_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_517_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_518_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_519_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_520_combine__common__factor,axiom,
! [A: int,E2: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_521_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_522_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_523_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_524_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_525_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_526_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_527_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_528_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_529_subset__insertI2,axiom,
! [A2: set_list_a,B3: set_list_a,B: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_530_subset__insertI2,axiom,
! [A2: set_a,B3: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_531_subset__insertI,axiom,
! [B3: set_list_a,A: list_a] : ( ord_le8861187494160871172list_a @ B3 @ ( insert_list_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_532_subset__insertI,axiom,
! [B3: set_a,A: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_533_subset__insert,axiom,
! [X: nat > list_a,A2: set_nat_list_a,B3: set_nat_list_a] :
( ~ ( member_nat_list_a @ X @ A2 )
=> ( ( ord_le2145805922479659755list_a @ A2 @ ( insert_nat_list_a @ X @ B3 ) )
= ( ord_le2145805922479659755list_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_534_subset__insert,axiom,
! [X: nat > a,A2: set_nat_a,B3: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ( ord_le871467723717165285_nat_a @ A2 @ ( insert_nat_a @ X @ B3 ) )
= ( ord_le871467723717165285_nat_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_535_subset__insert,axiom,
! [X: list_a,A2: set_list_a,B3: set_list_a] :
( ~ ( member_list_a @ X @ A2 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ B3 ) )
= ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_536_subset__insert,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_537_insert__mono,axiom,
! [C3: set_list_a,D: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ C3 @ D )
=> ( ord_le8861187494160871172list_a @ ( insert_list_a @ A @ C3 ) @ ( insert_list_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_538_insert__mono,axiom,
! [C3: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C3 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_539_singleton__inject,axiom,
! [A: list_a,B: list_a] :
( ( ( insert_list_a @ A @ bot_bot_set_list_a )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_540_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_541_insert__not__empty,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ A2 )
!= bot_bot_set_list_a ) ).
% insert_not_empty
thf(fact_542_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_543_doubleton__eq__iff,axiom,
! [A: list_a,B: list_a,C: list_a,D2: list_a] :
( ( ( insert_list_a @ A @ ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( insert_list_a @ C @ ( insert_list_a @ D2 @ bot_bot_set_list_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_544_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_545_singleton__iff,axiom,
! [B: nat > list_a,A: nat > list_a] :
( ( member_nat_list_a @ B @ ( insert_nat_list_a @ A @ bot_bo3806784159821827511list_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_546_singleton__iff,axiom,
! [B: nat > a,A: nat > a] :
( ( member_nat_a @ B @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_547_singleton__iff,axiom,
! [B: list_a,A: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_548_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_549_singletonD,axiom,
! [B: nat > list_a,A: nat > list_a] :
( ( member_nat_list_a @ B @ ( insert_nat_list_a @ A @ bot_bo3806784159821827511list_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_550_singletonD,axiom,
! [B: nat > a,A: nat > a] :
( ( member_nat_a @ B @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_551_singletonD,axiom,
! [B: list_a,A: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_552_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_553_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_554_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_555_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_556_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_557_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_558_eq__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D2 ) ) ) ).
% eq_add_iff2
thf(fact_559_eq__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D2: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
= D2 ) ) ).
% eq_add_iff1
thf(fact_560_subset__singletonD,axiom,
! [A2: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) )
=> ( ( A2 = bot_bot_set_list_a )
| ( A2
= ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ) ).
% subset_singletonD
thf(fact_561_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_562_subset__singleton__iff,axiom,
! [X5: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ X5 @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( ( X5 = bot_bot_set_list_a )
| ( X5
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_563_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_564_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D2 ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_565_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D2 ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D2 ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_566_ring__primeE_I1_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( P2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_567_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_568_x_Omaximalideal__prime,axiom,
! [I2: set_list_a] :
( ( maxima6585700282301356660t_unit @ I2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( primei6309817859076077608t_unit @ I2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.maximalideal_prime
thf(fact_569_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_570_x_Ocarrier__is__subalgebra,axiom,
! [K2: set_list_a] :
( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( embedd1768981623711841426t_unit @ K2 @ ( 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_571_x_Osubalgebra__in__carrier,axiom,
! [K2: set_list_a,V: set_list_a] :
( ( embedd1768981623711841426t_unit @ K2 @ 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_572_zeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroprimeideal
thf(fact_573_monoid__cancelI,axiom,
( ! [A3: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A3 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B2 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A3 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B2 @ C2 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_574_maximalideal__prime,axiom,
! [I2: set_a] :
( ( maximalideal_a_b @ I2 @ r )
=> ( primeideal_a_b @ I2 @ r ) ) ).
% maximalideal_prime
thf(fact_575_primeness__condition,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P2 )
= ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% primeness_condition
thf(fact_576_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_577_x_Ounit__factor,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.unit_factor
thf(fact_578_x_Oprod__unit__r,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.prod_unit_r
thf(fact_579_x_Oprod__unit__l,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.prod_unit_l
thf(fact_580_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_581_x_Oideal__eq__carrier__iff,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) )
= ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.ideal_eq_carrier_iff
thf(fact_582_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_583_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_584_assms_I1_J,axiom,
finite_finite_a @ s ).
% assms(1)
thf(fact_585_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_586_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_587_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_588_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_589_primeideal_Oprimeideal,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( primei6309817859076077608t_unit @ I2 @ R )
=> ( primei6309817859076077608t_unit @ I2 @ R ) ) ).
% primeideal.primeideal
thf(fact_590_primeideal_Oprimeideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( primeideal_a_b @ I2 @ R )
=> ( primeideal_a_b @ I2 @ R ) ) ).
% primeideal.primeideal
thf(fact_591_primeideal_OI__notcarr,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( primei6309817859076077608t_unit @ I2 @ R )
=> ( ( partia5361259788508890537t_unit @ R )
!= I2 ) ) ).
% primeideal.I_notcarr
thf(fact_592_primeideal_OI__notcarr,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( primeideal_a_b @ I2 @ R )
=> ( ( partia707051561876973205xt_a_b @ R )
!= I2 ) ) ).
% primeideal.I_notcarr
thf(fact_593_primeideal_OI__prime,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( primei6309817859076077608t_unit @ I2 @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ A @ B ) @ I2 )
=> ( ( member_list_a @ A @ I2 )
| ( member_list_a @ B @ I2 ) ) ) ) ) ) ).
% primeideal.I_prime
thf(fact_594_primeideal_OI__prime,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( primeideal_a_b @ I2 @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ R @ A @ B ) @ I2 )
=> ( ( member_a @ A @ I2 )
| ( member_a @ B @ I2 ) ) ) ) ) ) ).
% primeideal.I_prime
thf(fact_595_Ring_Ofield__Units,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( units_2471184348132832486t_unit @ R )
= ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) ) ) ).
% Ring.field_Units
thf(fact_596_Ring_Ofield__Units,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( ( units_5837875185506529638t_unit @ R )
= ( minus_4782336368215558443list_a @ ( partia141011252114345353t_unit @ R ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) ) ) ).
% Ring.field_Units
thf(fact_597_Ring_Ofield__Units,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( units_2932844235741507942t_unit @ R )
= ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ) ).
% Ring.field_Units
thf(fact_598_Ring_Ofield__Units,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( units_a_ring_ext_a_b @ R )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ).
% Ring.field_Units
thf(fact_599_p__def,axiom,
( p
= ( lagran9092808442999052491ux_a_b @ r @ s ) ) ).
% p_def
thf(fact_600_ring__primeI,axiom,
! [P2: a] :
( ( P2
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P2 )
=> ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% ring_primeI
thf(fact_601_ring__primeE_I3_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( prime_a_ring_ext_a_b @ r @ P2 ) ) ) ).
% ring_primeE(3)
thf(fact_602_primeideal__iff__prime,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( 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 @ P2 ) @ r )
= ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% primeideal_iff_prime
thf(fact_603_x_Osubfield__m__inv__simprule,axiom,
! [K2: set_list_a,K: list_a,A: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K @ ( minus_646659088055828811list_a @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A ) @ K2 )
=> ( member_list_a @ A @ K2 ) ) ) ) ) ).
% x.subfield_m_inv_simprule
thf(fact_604_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_605_prod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_606_prod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_607_unit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_608_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_609_ideal__eq__carrier__iff,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( partia707051561876973205xt_a_b @ r )
= ( cgenid547466209912283029xt_a_b @ r @ A ) )
= ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ideal_eq_carrier_iff
thf(fact_610_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_611_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_612_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_613_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_614_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_615_ring__irreducibleE_I5_J,axiom,
! [R2: a,A: a,B: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_616_x_Osubring__props_I2_J,axiom,
! [K2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K2 ) ) ).
% x.subring_props(2)
thf(fact_617_x_Osubring__props_I7_J,axiom,
! [K2: set_list_a,H12: list_a,H22: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H12 @ K2 )
=> ( ( member_list_a @ H22 @ K2 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H12 @ H22 ) @ K2 ) ) ) ) ).
% x.subring_props(7)
thf(fact_618_x_Osubring__props_I6_J,axiom,
! [K2: set_list_a,H12: list_a,H22: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H12 @ K2 )
=> ( ( member_list_a @ H22 @ K2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H12 @ H22 ) @ K2 ) ) ) ) ).
% x.subring_props(6)
thf(fact_619_cgenideal__is__principalideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I ) @ r ) ) ).
% cgenideal_is_principalideal
thf(fact_620_x_Osubring__props_I4_J,axiom,
! [K2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( K2 != bot_bot_set_list_a ) ) ).
% x.subring_props(4)
thf(fact_621_x_Osubring__props_I3_J,axiom,
! [K2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K2 ) ) ).
% x.subring_props(3)
thf(fact_622_x_Osubring__props_I1_J,axiom,
! [K2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subring_props(1)
thf(fact_623_irreducible__imp__maximalideal,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P2 )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P2 ) @ r ) ) ) ).
% irreducible_imp_maximalideal
thf(fact_624_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_625_cgenideal__eq__genideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( cgenid547466209912283029xt_a_b @ r @ I )
= ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% cgenideal_eq_genideal
thf(fact_626_lagrange__aux__poly,axiom,
! [S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ( ord_less_eq_set_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( lagran9092808442999052491ux_a_b @ r @ S2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% lagrange_aux_poly
thf(fact_627_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_628_x_Oline__extension__smult__closed,axiom,
! [K2: set_list_a,E: set_list_a,A: list_a,K: list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [K4: list_a,V2: list_a] :
( ( member_list_a @ K4 @ K2 )
=> ( ( member_list_a @ V2 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ V2 ) @ E ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K @ K2 )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A @ E ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ U ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A @ E ) ) ) ) ) ) ) ) ).
% x.line_extension_smult_closed
thf(fact_629_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 ) )
=> ( ! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ r @ R2 ) ) ) ) ).
% ring_irreducibleI
thf(fact_630_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_631_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_632_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_633_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_634_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_635_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
lagran9092808442999052491ux_a_b = lagran9092808442999052491ux_a_b ).
% ring.lagrange_basis_polynomial_aux.cong
thf(fact_636_field__iff__prime,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( field_6045675692312731021t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) ) )
= ( ring_ring_prime_a_b @ r @ A ) ) ) ).
% field_iff_prime
thf(fact_637_finite__Diff__insert,axiom,
! [A2: set_list_a,A: list_a,B3: set_list_a] :
( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B3 ) ) )
= ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) ) ) ).
% finite_Diff_insert
thf(fact_638_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B3: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B3 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% finite_Diff_insert
thf(fact_639_x_Osubfield__m__inv_I2_J,axiom,
! [K2: set_list_a,K: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K @ ( minus_646659088055828811list_a @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.subfield_m_inv(2)
thf(fact_640_x_Osubfield__m__inv_I3_J,axiom,
! [K2: set_list_a,K: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K @ ( minus_646659088055828811list_a @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ K )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.subfield_m_inv(3)
thf(fact_641_x_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K2: set_list_a,E: set_list_a,V: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ E )
=> ( ( embedd1768981623711841426t_unit @ K2 @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ V @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ V ) ) ) ) ) ).
% x.subalbegra_incl_imp_finite_dimension
thf(fact_642_x_Otelescopic__base__dim_I1_J,axiom,
! [K2: set_list_a,F2: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subfie1779122896746047282t_unit @ F2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ F2 )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F2 @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ E ) ) ) ) ) ).
% x.telescopic_base_dim(1)
thf(fact_643_x_Oinv__eq__imp__eq,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 ) ) ) )
=> ( ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) )
=> ( X = Y ) ) ) ) ).
% x.inv_eq_imp_eq
thf(fact_644_x_Oinv__eq__one__eq,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( X
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.inv_eq_one_eq
thf(fact_645_x_Ofinite__dimension__imp__subalgebra,axiom,
! [K2: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ E )
=> ( embedd1768981623711841426t_unit @ K2 @ E @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.finite_dimension_imp_subalgebra
thf(fact_646_x_Oinv__unique_H,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 )
= ( one_li8328186300101108157t_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 ) ) ) )
=> ( Y
= ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) ) ) ) ) ) ).
% x.inv_unique'
thf(fact_647_x_Oinv__char,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 )
= ( one_li8328186300101108157t_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 ) ) ) )
=> ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= Y ) ) ) ) ) ).
% x.inv_char
thf(fact_648_x_Ocomm__inv__char,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 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= Y ) ) ) ) ).
% x.comm_inv_char
thf(fact_649_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_650_finite__insert,axiom,
! [A: list_a,A2: set_list_a] :
( ( finite_finite_list_a @ ( insert_list_a @ A @ A2 ) )
= ( finite_finite_list_a @ A2 ) ) ).
% finite_insert
thf(fact_651_x_Osubfield__m__inv_I1_J,axiom,
! [K2: set_list_a,K: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K @ ( minus_646659088055828811list_a @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( minus_646659088055828811list_a @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ) ).
% x.subfield_m_inv(1)
thf(fact_652_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_653_x_OUnits__inv__inv,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
= X ) ) ).
% x.Units_inv_inv
thf(fact_654_x_OUnits__inv__Units,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_inv_Units
thf(fact_655_x_Oinv__one,axiom,
( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.inv_one
thf(fact_656_x_OUnits__inv__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_inv_closed
thf(fact_657_x_OUnits__r__inv,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_r_inv
thf(fact_658_x_OUnits__l__inv,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_l_inv
thf(fact_659_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_660_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_661_finite__has__minimal2,axiom,
! [A2: set_set_list_a,A: set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( member_set_list_a @ A @ A2 )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A2 )
& ( ord_le8861187494160871172list_a @ X2 @ A )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A2 )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_662_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_663_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_664_finite__has__maximal2,axiom,
! [A2: set_set_list_a,A: set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( member_set_list_a @ A @ A2 )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A2 )
& ( ord_le8861187494160871172list_a @ A @ X2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A2 )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_665_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ A @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_666_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_667_rev__finite__subset,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B3 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( finite_finite_list_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_668_rev__finite__subset,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_669_infinite__super,axiom,
! [S2: set_list_a,T2: set_list_a] :
( ( ord_le8861187494160871172list_a @ S2 @ T2 )
=> ( ~ ( finite_finite_list_a @ S2 )
=> ~ ( finite_finite_list_a @ T2 ) ) ) ).
% infinite_super
thf(fact_670_infinite__super,axiom,
! [S2: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S2 @ T2 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_671_finite__subset,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( finite_finite_list_a @ B3 )
=> ( finite_finite_list_a @ A2 ) ) ) ).
% finite_subset
thf(fact_672_finite__subset,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( finite_finite_a @ B3 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_673_infinite__imp__nonempty,axiom,
! [S2: set_list_a] :
( ~ ( finite_finite_list_a @ S2 )
=> ( S2 != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_674_infinite__imp__nonempty,axiom,
! [S2: set_a] :
( ~ ( finite_finite_a @ S2 )
=> ( S2 != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_675_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_676_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_677_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_678_finite_OinsertI,axiom,
! [A2: set_list_a,A: list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite_finite_list_a @ ( insert_list_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_679_finite__has__minimal,axiom,
! [A2: set_set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( A2 != bot_bo3186585308812441520list_a )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A2 )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_680_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_681_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_682_finite__has__maximal,axiom,
! [A2: set_set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( A2 != bot_bo3186585308812441520list_a )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A2 )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_683_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_684_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_685_infinite__finite__induct,axiom,
! [P: set_nat_list_a > $o,A2: set_nat_list_a] :
( ! [A5: set_nat_list_a] :
( ~ ( finite7630042315537210004list_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bo3806784159821827511list_a )
=> ( ! [X2: nat > list_a,F3: set_nat_list_a] :
( ( finite7630042315537210004list_a @ F3 )
=> ( ~ ( member_nat_list_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_list_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_686_infinite__finite__induct,axiom,
! [P: set_nat_a > $o,A2: set_nat_a] :
( ! [A5: set_nat_a] :
( ~ ( finite_finite_nat_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [X2: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ~ ( member_nat_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_687_infinite__finite__induct,axiom,
! [P: set_list_a > $o,A2: set_list_a] :
( ! [A5: set_list_a] :
( ~ ( finite_finite_list_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X2: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ~ ( member_list_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_688_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A5: set_a] :
( ~ ( finite_finite_a @ A5 )
=> ( P @ A5 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_689_finite__ne__induct,axiom,
! [F2: set_nat_list_a,P: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ F2 )
=> ( ( F2 != bot_bo3806784159821827511list_a )
=> ( ! [X2: nat > list_a] : ( P @ ( insert_nat_list_a @ X2 @ bot_bo3806784159821827511list_a ) )
=> ( ! [X2: nat > list_a,F3: set_nat_list_a] :
( ( finite7630042315537210004list_a @ F3 )
=> ( ( F3 != bot_bo3806784159821827511list_a )
=> ( ~ ( member_nat_list_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_list_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_690_finite__ne__induct,axiom,
! [F2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ F2 )
=> ( ( F2 != bot_bot_set_nat_a )
=> ( ! [X2: nat > a] : ( P @ ( insert_nat_a @ X2 @ bot_bot_set_nat_a ) )
=> ( ! [X2: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( F3 != bot_bot_set_nat_a )
=> ( ~ ( member_nat_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_691_finite__ne__induct,axiom,
! [F2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( F2 != bot_bot_set_list_a )
=> ( ! [X2: list_a] : ( P @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
=> ( ! [X2: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( F3 != bot_bot_set_list_a )
=> ( ~ ( member_list_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_692_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_693_finite__induct,axiom,
! [F2: set_nat_list_a,P: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ F2 )
=> ( ( P @ bot_bo3806784159821827511list_a )
=> ( ! [X2: nat > list_a,F3: set_nat_list_a] :
( ( finite7630042315537210004list_a @ F3 )
=> ( ~ ( member_nat_list_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_list_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_694_finite__induct,axiom,
! [F2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ F2 )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [X2: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ~ ( member_nat_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_695_finite__induct,axiom,
! [F2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X2: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ~ ( member_list_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_696_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_697_finite_Osimps,axiom,
( finite_finite_list_a
= ( ^ [A6: set_list_a] :
( ( A6 = bot_bot_set_list_a )
| ? [A4: set_list_a,B6: list_a] :
( ( A6
= ( insert_list_a @ B6 @ A4 ) )
& ( finite_finite_list_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_698_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A6: set_a] :
( ( A6 = bot_bot_set_a )
| ? [A4: set_a,B6: a] :
( ( A6
= ( insert_a @ B6 @ A4 ) )
& ( finite_finite_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_699_finite_Ocases,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( ( A != bot_bot_set_list_a )
=> ~ ! [A5: set_list_a] :
( ? [A3: list_a] :
( A
= ( insert_list_a @ A3 @ A5 ) )
=> ~ ( finite_finite_list_a @ A5 ) ) ) ) ).
% finite.cases
thf(fact_700_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A5: set_a] :
( ? [A3: a] :
( A
= ( insert_a @ A3 @ A5 ) )
=> ~ ( finite_finite_a @ A5 ) ) ) ) ).
% finite.cases
thf(fact_701_finite__subset__induct_H,axiom,
! [F2: set_nat_list_a,A2: set_nat_list_a,P: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ F2 )
=> ( ( ord_le2145805922479659755list_a @ F2 @ A2 )
=> ( ( P @ bot_bo3806784159821827511list_a )
=> ( ! [A3: nat > list_a,F3: set_nat_list_a] :
( ( finite7630042315537210004list_a @ F3 )
=> ( ( member_nat_list_a @ A3 @ A2 )
=> ( ( ord_le2145805922479659755list_a @ F3 @ A2 )
=> ( ~ ( member_nat_list_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_list_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_702_finite__subset__induct_H,axiom,
! [F2: set_nat_a,A2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ F2 )
=> ( ( ord_le871467723717165285_nat_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [A3: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( member_nat_a @ A3 @ A2 )
=> ( ( ord_le871467723717165285_nat_a @ F3 @ A2 )
=> ( ~ ( member_nat_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_703_finite__subset__induct_H,axiom,
! [F2: set_list_a,A2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( ord_le8861187494160871172list_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A3: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( member_list_a @ A3 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ F3 @ A2 )
=> ( ~ ( member_list_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_704_finite__subset__induct_H,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_705_finite__subset__induct,axiom,
! [F2: set_nat_list_a,A2: set_nat_list_a,P: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ F2 )
=> ( ( ord_le2145805922479659755list_a @ F2 @ A2 )
=> ( ( P @ bot_bo3806784159821827511list_a )
=> ( ! [A3: nat > list_a,F3: set_nat_list_a] :
( ( finite7630042315537210004list_a @ F3 )
=> ( ( member_nat_list_a @ A3 @ A2 )
=> ( ~ ( member_nat_list_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_list_a @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_706_finite__subset__induct,axiom,
! [F2: set_nat_a,A2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ F2 )
=> ( ( ord_le871467723717165285_nat_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [A3: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( member_nat_a @ A3 @ A2 )
=> ( ~ ( member_nat_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_707_finite__subset__induct,axiom,
! [F2: set_list_a,A2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( ord_le8861187494160871172list_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A3: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( member_list_a @ A3 @ A2 )
=> ( ~ ( member_list_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_708_finite__subset__induct,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A3 @ A2 )
=> ( ~ ( member_a @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_709_finite__empty__induct,axiom,
! [A2: set_nat_list_a,P: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: nat > list_a,A5: set_nat_list_a] :
( ( finite7630042315537210004list_a @ A5 )
=> ( ( member_nat_list_a @ A3 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_4169782841487898290list_a @ A5 @ ( insert_nat_list_a @ A3 @ bot_bo3806784159821827511list_a ) ) ) ) ) )
=> ( P @ bot_bo3806784159821827511list_a ) ) ) ) ).
% finite_empty_induct
thf(fact_710_finite__empty__induct,axiom,
! [A2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: nat > a,A5: set_nat_a] :
( ( finite_finite_nat_a @ A5 )
=> ( ( member_nat_a @ A3 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_490503922182417452_nat_a @ A5 @ ( insert_nat_a @ A3 @ bot_bot_set_nat_a ) ) ) ) ) )
=> ( P @ bot_bot_set_nat_a ) ) ) ) ).
% finite_empty_induct
thf(fact_711_finite__empty__induct,axiom,
! [A2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: list_a,A5: set_list_a] :
( ( finite_finite_list_a @ A5 )
=> ( ( member_list_a @ A3 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_646659088055828811list_a @ A5 @ ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) ) ) ) )
=> ( P @ bot_bot_set_list_a ) ) ) ) ).
% finite_empty_induct
thf(fact_712_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A3: a,A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( member_a @ A3 @ A5 )
=> ( ( P @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_713_infinite__coinduct,axiom,
! [X5: set_list_a > $o,A2: set_list_a] :
( ( X5 @ A2 )
=> ( ! [A5: set_list_a] :
( ( X5 @ A5 )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ A5 )
& ( ( X5 @ ( minus_646659088055828811list_a @ A5 @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) )
| ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A5 @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) ) ) ) )
=> ~ ( finite_finite_list_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_714_infinite__coinduct,axiom,
! [X5: set_a > $o,A2: set_a] :
( ( X5 @ A2 )
=> ( ! [A5: set_a] :
( ( X5 @ A5 )
=> ? [X4: a] :
( ( member_a @ X4 @ A5 )
& ( ( X5 @ ( minus_minus_set_a @ A5 @ ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A5 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_715_infinite__remove,axiom,
! [S2: set_list_a,A: list_a] :
( ~ ( finite_finite_list_a @ S2 )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% infinite_remove
thf(fact_716_infinite__remove,axiom,
! [S2: set_a,A: a] :
( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_717_remove__induct,axiom,
! [P: set_nat_list_a > $o,B3: set_nat_list_a] :
( ( P @ bot_bo3806784159821827511list_a )
=> ( ( ~ ( finite7630042315537210004list_a @ B3 )
=> ( P @ B3 ) )
=> ( ! [A5: set_nat_list_a] :
( ( finite7630042315537210004list_a @ A5 )
=> ( ( A5 != bot_bo3806784159821827511list_a )
=> ( ( ord_le2145805922479659755list_a @ A5 @ B3 )
=> ( ! [X4: nat > list_a] :
( ( member_nat_list_a @ X4 @ A5 )
=> ( P @ ( minus_4169782841487898290list_a @ A5 @ ( insert_nat_list_a @ X4 @ bot_bo3806784159821827511list_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_718_remove__induct,axiom,
! [P: set_nat_a > $o,B3: set_nat_a] :
( ( P @ bot_bot_set_nat_a )
=> ( ( ~ ( finite_finite_nat_a @ B3 )
=> ( P @ B3 ) )
=> ( ! [A5: set_nat_a] :
( ( finite_finite_nat_a @ A5 )
=> ( ( A5 != bot_bot_set_nat_a )
=> ( ( ord_le871467723717165285_nat_a @ A5 @ B3 )
=> ( ! [X4: nat > a] :
( ( member_nat_a @ X4 @ A5 )
=> ( P @ ( minus_490503922182417452_nat_a @ A5 @ ( insert_nat_a @ X4 @ bot_bot_set_nat_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_719_remove__induct,axiom,
! [P: set_list_a > $o,B3: set_list_a] :
( ( P @ bot_bot_set_list_a )
=> ( ( ~ ( finite_finite_list_a @ B3 )
=> ( P @ B3 ) )
=> ( ! [A5: set_list_a] :
( ( finite_finite_list_a @ A5 )
=> ( ( A5 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A5 @ B3 )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ A5 )
=> ( P @ ( minus_646659088055828811list_a @ A5 @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_720_remove__induct,axiom,
! [P: set_a > $o,B3: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B3 )
=> ( P @ B3 ) )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A5 @ B3 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_721_finite__remove__induct,axiom,
! [B3: set_nat_list_a,P: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ B3 )
=> ( ( P @ bot_bo3806784159821827511list_a )
=> ( ! [A5: set_nat_list_a] :
( ( finite7630042315537210004list_a @ A5 )
=> ( ( A5 != bot_bo3806784159821827511list_a )
=> ( ( ord_le2145805922479659755list_a @ A5 @ B3 )
=> ( ! [X4: nat > list_a] :
( ( member_nat_list_a @ X4 @ A5 )
=> ( P @ ( minus_4169782841487898290list_a @ A5 @ ( insert_nat_list_a @ X4 @ bot_bo3806784159821827511list_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_722_finite__remove__induct,axiom,
! [B3: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ B3 )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [A5: set_nat_a] :
( ( finite_finite_nat_a @ A5 )
=> ( ( A5 != bot_bot_set_nat_a )
=> ( ( ord_le871467723717165285_nat_a @ A5 @ B3 )
=> ( ! [X4: nat > a] :
( ( member_nat_a @ X4 @ A5 )
=> ( P @ ( minus_490503922182417452_nat_a @ A5 @ ( insert_nat_a @ X4 @ bot_bot_set_nat_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_723_finite__remove__induct,axiom,
! [B3: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ B3 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A5: set_list_a] :
( ( finite_finite_list_a @ A5 )
=> ( ( A5 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A5 @ B3 )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ A5 )
=> ( P @ ( minus_646659088055828811list_a @ A5 @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_724_finite__remove__induct,axiom,
! [B3: set_a,P: set_a > $o] :
( ( finite_finite_a @ B3 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ( A5 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A5 @ B3 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( P @ ( minus_minus_set_a @ A5 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A5 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_725_principal__domain_Ofield__iff__prime,axiom,
! [R: partia2175431115845679010xt_a_b,A: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( field_6045675692312731021t_unit @ ( factRing_a_b @ R @ ( cgenid547466209912283029xt_a_b @ R @ A ) ) )
= ( ring_ring_prime_a_b @ R @ A ) ) ) ) ).
% principal_domain.field_iff_prime
thf(fact_726_principal__domain_Ofield__iff__prime,axiom,
! [R: partia2670972154091845814t_unit,A: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ A @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( field_26233345952514695t_unit @ ( factRi3329376332477095402t_unit @ R @ ( cgenid9131348535277946915t_unit @ R @ A ) ) )
= ( ring_r6430282645014804837t_unit @ R @ A ) ) ) ) ).
% principal_domain.field_iff_prime
thf(fact_727_domain__iff__prime,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) ) )
= ( ring_ring_prime_a_b @ r @ A ) ) ) ).
% domain_iff_prime
thf(fact_728_x_Ospace__subgroup__props_I6_J,axiom,
! [K2: set_list_a,N: nat,E: set_list_a,K: list_a,A: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K2 @ E )
=> ( ( member_list_a @ K @ ( minus_646659088055828811list_a @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A ) @ E )
=> ( member_list_a @ A @ E ) ) ) ) ) ) ).
% x.space_subgroup_props(6)
thf(fact_729_x_Oirreducible__prod__rI,axiom,
! [A: list_a,B: list_a] :
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ) ).
% x.irreducible_prod_rI
thf(fact_730_x_Oirreducible__prod__lI,axiom,
! [B: list_a,A: list_a] :
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ) ).
% x.irreducible_prod_lI
thf(fact_731_x_Odimension__is__inj,axiom,
! [K2: set_list_a,N: nat,E: set_list_a,M: nat] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K2 @ E )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M @ K2 @ E )
=> ( N = M ) ) ) ) ).
% x.dimension_is_inj
thf(fact_732_x_Ofinite__dimensionE_H,axiom,
! [K2: set_list_a,E: set_list_a] :
( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ E )
=> ~ ! [N2: nat] :
~ ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N2 @ K2 @ E ) ) ).
% x.finite_dimensionE'
thf(fact_733_x_Ofinite__dimensionI,axiom,
! [N: nat,K2: set_list_a,E: set_list_a] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K2 @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ E ) ) ).
% x.finite_dimensionI
thf(fact_734_x_Ofinite__dimension__def,axiom,
! [K2: set_list_a,E: set_list_a] :
( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ E )
= ( ? [N4: nat] : ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N4 @ K2 @ E ) ) ) ).
% x.finite_dimension_def
thf(fact_735_x_Ospace__subgroup__props_I2_J,axiom,
! [K2: set_list_a,N: nat,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K2 @ E )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ E ) ) ) ).
% x.space_subgroup_props(2)
thf(fact_736_x_Ospace__subgroup__props_I3_J,axiom,
! [K2: set_list_a,N: nat,E: set_list_a,V1: list_a,V22: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K2 @ E )
=> ( ( member_list_a @ V1 @ E )
=> ( ( member_list_a @ V22 @ E )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ V1 @ V22 ) @ E ) ) ) ) ) ).
% x.space_subgroup_props(3)
thf(fact_737_x_Otelescopic__base,axiom,
! [K2: set_list_a,F2: set_list_a,N: nat,M: nat,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subfie1779122896746047282t_unit @ F2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K2 @ F2 )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M @ F2 @ E )
=> ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( times_times_nat @ N @ M ) @ K2 @ E ) ) ) ) ) ).
% x.telescopic_base
thf(fact_738_x_Ospace__subgroup__props_I5_J,axiom,
! [K2: set_list_a,N: nat,E: set_list_a,K: list_a,V3: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K2 @ E )
=> ( ( member_list_a @ K @ K2 )
=> ( ( member_list_a @ V3 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ V3 ) @ E ) ) ) ) ) ).
% x.space_subgroup_props(5)
thf(fact_739_x_Ounique__dimension,axiom,
! [K2: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ E )
=> ? [X2: nat] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ K2 @ E )
& ! [Y5: nat] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y5 @ K2 @ E )
=> ( Y5 = X2 ) ) ) ) ) ).
% x.unique_dimension
thf(fact_740_x_Ospace__subgroup__props_I1_J,axiom,
! [K2: set_list_a,N: nat,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K2 @ E )
=> ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.space_subgroup_props(1)
thf(fact_741_x_OSuc__dim,axiom,
! [V3: list_a,E: set_list_a,N: nat,K2: set_list_a] :
( ( member_list_a @ V3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ~ ( member_list_a @ V3 @ E )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K2 @ E )
=> ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( suc @ N ) @ K2 @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ V3 @ E ) ) ) ) ) ).
% x.Suc_dim
thf(fact_742_x_Odimension__backwards,axiom,
! [K2: set_list_a,N: nat,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( suc @ N ) @ K2 @ E )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ? [E3: set_list_a] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ K2 @ E3 )
& ~ ( member_list_a @ X2 @ E3 )
& ( E
= ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ X2 @ E3 ) ) ) ) ) ) ).
% x.dimension_backwards
thf(fact_743_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_744_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_745_domain_Ozero__is__irreducible__iff__field,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( irredu943254396193320253t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) )
= ( field_26233345952514695t_unit @ R ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_746_domain_Ozero__is__irreducible__iff__field,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( irredu5346329325703585725t_unit @ R @ ( zero_s2174465271003423091t_unit @ R ) )
= ( field_6045675692312731021t_unit @ R ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_747_domain_Ozero__is__irreducible__iff__field,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( irredu4230924414530676029t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) )
= ( field_6388047844668329575t_unit @ R ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_748_domain_Ozero__is__irreducible__iff__field,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( irredu6211895646901577903xt_a_b @ R @ ( zero_a_b @ R ) )
= ( field_a_b @ R ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_749_field_Oaxioms_I1_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( domain1617769409708967785t_unit @ R ) ) ).
% field.axioms(1)
thf(fact_750_field_Oaxioms_I1_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( domain4236798911309298543t_unit @ R ) ) ).
% field.axioms(1)
thf(fact_751_field_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% field.axioms(1)
thf(fact_752_field_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% field.axioms(1)
thf(fact_753_domain_Oring__irreducibleE_I2_J,axiom,
! [R: partia6043505979758434576t_unit,R2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ R2 )
=> ( irredu5346329325703585725t_unit @ R @ R2 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_754_domain_Oring__irreducibleE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ( irredu6211895646901577903xt_a_b @ R @ R2 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_755_domain_Oring__irreducibleE_I2_J,axiom,
! [R: partia2670972154091845814t_unit,R2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R2 )
=> ( irredu4230924414530676029t_unit @ R @ R2 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_756_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_p2862007038493914190t_unit @ R )
=> ( domain4236798911309298543t_unit @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_757_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_758_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_759_noetherian__domain_Oaxioms_I2_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_n3212398840814694743t_unit @ R )
=> ( domain4236798911309298543t_unit @ R ) ) ).
% noetherian_domain.axioms(2)
thf(fact_760_noetherian__domain_Oaxioms_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_n4705423059119889713t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% noetherian_domain.axioms(2)
thf(fact_761_noetherian__domain_Oaxioms_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_n4045954140777738665in_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% noetherian_domain.axioms(2)
thf(fact_762_factorial__domain_Oaxioms_I1_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_f6820247627256571077t_unit @ R )
=> ( domain4236798911309298543t_unit @ R ) ) ).
% factorial_domain.axioms(1)
thf(fact_763_factorial__domain_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_f796907574329358751t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% factorial_domain.axioms(1)
thf(fact_764_factorial__domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_f5272581269873410839in_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% factorial_domain.axioms(1)
thf(fact_765_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( prime_4522187476880896870t_unit @ R @ ( zero_s2174465271003423091t_unit @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_766_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( prime_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_767_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( prime_2011924034616061926t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_768_ring__irreducible__def,axiom,
( ring_r999134135267193926le_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,A6: a] :
( ( A6
!= ( zero_a_b @ R3 ) )
& ( irredu6211895646901577903xt_a_b @ R3 @ A6 ) ) ) ) ).
% ring_irreducible_def
thf(fact_769_ring__irreducible__def,axiom,
( ring_r932985474545269838t_unit
= ( ^ [R3: partia2670972154091845814t_unit,A6: list_a] :
( ( A6
!= ( zero_l4142658623432671053t_unit @ R3 ) )
& ( irredu4230924414530676029t_unit @ R3 @ A6 ) ) ) ) ).
% ring_irreducible_def
thf(fact_770_noetherian__domain__def,axiom,
( ring_n3212398840814694743t_unit
= ( ^ [R3: partia6043505979758434576t_unit] :
( ( ring_n5014428767265248323t_unit @ R3 )
& ( domain4236798911309298543t_unit @ R3 ) ) ) ) ).
% noetherian_domain_def
thf(fact_771_noetherian__domain__def,axiom,
( ring_n4705423059119889713t_unit
= ( ^ [R3: partia2670972154091845814t_unit] :
( ( ring_n5188127996776581661t_unit @ R3 )
& ( domain6553523120543210313t_unit @ R3 ) ) ) ) ).
% noetherian_domain_def
thf(fact_772_noetherian__domain__def,axiom,
( ring_n4045954140777738665in_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b] :
( ( ring_n3639167112692572309ng_a_b @ R3 )
& ( domain_a_b @ R3 ) ) ) ) ).
% noetherian_domain_def
thf(fact_773_noetherian__domain_Ointro,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_n5014428767265248323t_unit @ R )
=> ( ( domain4236798911309298543t_unit @ R )
=> ( ring_n3212398840814694743t_unit @ R ) ) ) ).
% noetherian_domain.intro
thf(fact_774_noetherian__domain_Ointro,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_n5188127996776581661t_unit @ R )
=> ( ( domain6553523120543210313t_unit @ R )
=> ( ring_n4705423059119889713t_unit @ R ) ) ) ).
% noetherian_domain.intro
thf(fact_775_noetherian__domain_Ointro,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_n3639167112692572309ng_a_b @ R )
=> ( ( domain_a_b @ R )
=> ( ring_n4045954140777738665in_a_b @ R ) ) ) ).
% noetherian_domain.intro
thf(fact_776_domain_Oone__not__zero,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( one_se211549098623999037t_unit @ R )
!= ( zero_s2174465271003423091t_unit @ R ) ) ) ).
% domain.one_not_zero
thf(fact_777_domain_Oone__not__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% domain.one_not_zero
thf(fact_778_domain_Oone__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% domain.one_not_zero
thf(fact_779_domain_Ozero__not__one,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( zero_s2174465271003423091t_unit @ R )
!= ( one_se211549098623999037t_unit @ R ) ) ) ).
% domain.zero_not_one
thf(fact_780_domain_Ozero__not__one,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( zero_l4142658623432671053t_unit @ R )
!= ( one_li8328186300101108157t_unit @ R ) ) ) ).
% domain.zero_not_one
thf(fact_781_domain_Ozero__not__one,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( zero_a_b @ R )
!= ( one_a_ring_ext_a_b @ R ) ) ) ).
% domain.zero_not_one
thf(fact_782_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia6043505979758434576t_unit,R2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ R2 )
=> ( R2
!= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_783_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ( R2
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_784_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,R2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R2 )
=> ( R2
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_785_domain_Oring__primeE_I1_J,axiom,
! [R: partia6043505979758434576t_unit,P2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ P2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r6795642478576035723t_unit @ R @ P2 )
=> ( P2
!= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_786_domain_Oring__primeE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r6430282645014804837t_unit @ R @ P2 )
=> ( P2
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_787_domain_Oring__primeE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P2 )
=> ( P2
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_788_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia6043505979758434576t_unit,R2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ R2 )
=> ~ ( member_set_a @ R2 @ ( units_2471184348132832486t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_789_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_790_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia2670972154091845814t_unit,R2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R2 )
=> ~ ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_791_domain_Oring__primeE_I3_J,axiom,
! [R: partia6043505979758434576t_unit,P2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ P2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r6795642478576035723t_unit @ R @ P2 )
=> ( prime_4522187476880896870t_unit @ R @ P2 ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_792_domain_Oring__primeE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,P2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r6430282645014804837t_unit @ R @ P2 )
=> ( prime_2011924034616061926t_unit @ R @ P2 ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_793_domain_Oring__primeE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P2 )
=> ( prime_a_ring_ext_a_b @ R @ P2 ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_794_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia6043505979758434576t_unit,R2: set_a,A: set_a,B: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ R2 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( R2
= ( mult_s7930653359683758801t_unit @ R @ A @ B ) )
=> ( ( member_set_a @ A @ ( units_2471184348132832486t_unit @ R ) )
| ( member_set_a @ B @ ( units_2471184348132832486t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_795_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ R @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ R ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_796_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2670972154091845814t_unit,R2: list_a,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( R2
= ( mult_l7073676228092353617t_unit @ R @ A @ B ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ R ) )
| ( member_list_a @ B @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_797_domain_Ointegral__iff,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ A @ B )
= ( zero_s2174465271003423091t_unit @ R ) )
= ( ( A
= ( zero_s2174465271003423091t_unit @ R ) )
| ( B
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_798_domain_Ointegral__iff,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
= ( ( A
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_799_domain_Ointegral__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
= ( ( A
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_800_domain_Om__rcancel,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a,C: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( A
!= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ B @ A )
= ( mult_s7930653359683758801t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_801_domain_Om__rcancel,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ B @ A )
= ( mult_l7073676228092353617t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_802_domain_Om__rcancel,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R )
=> ( ( A
!= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ B @ A )
= ( mult_a_ring_ext_a_b @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_803_domain_Om__lcancel,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a,C: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( A
!= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ A @ B )
= ( mult_s7930653359683758801t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_804_domain_Om__lcancel,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( mult_l7073676228092353617t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_805_domain_Om__lcancel,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R )
=> ( ( A
!= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( mult_a_ring_ext_a_b @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_806_domain_Ointegral,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ A @ B )
= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( A
= ( zero_s2174465271003423091t_unit @ R ) )
| ( B
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_807_domain_Ointegral,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( A
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_808_domain_Ointegral,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_809_domain_Ozeroprimeideal,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( primei7645216761534224334t_unit @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) @ R ) ) ).
% domain.zeroprimeideal
thf(fact_810_domain_Ozeroprimeideal,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R ) ) ).
% domain.zeroprimeideal
thf(fact_811_domain_Ozeroprimeideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ).
% domain.zeroprimeideal
thf(fact_812_domain_Oring__irreducibleI,axiom,
! [R: partia6043505979758434576t_unit,R2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R2 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ~ ( member_set_a @ R2 @ ( units_2471184348132832486t_unit @ R ) )
=> ( ! [A3: set_a,B2: set_a] :
( ( member_set_a @ A3 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( R2
= ( mult_s7930653359683758801t_unit @ R @ A3 @ B2 ) )
=> ( ( member_set_a @ A3 @ ( units_2471184348132832486t_unit @ R ) )
| ( member_set_a @ B2 @ ( units_2471184348132832486t_unit @ R ) ) ) ) ) )
=> ( ring_r7790391342995787508t_unit @ R @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleI
thf(fact_813_domain_Oring__irreducibleI,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( 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 ) )
=> ( ! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ R @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ R ) )
| ( member_a @ B2 @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ R @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleI
thf(fact_814_domain_Oring__irreducibleI,axiom,
! [R: partia2670972154091845814t_unit,R2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R2 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ~ ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ R ) )
=> ( ! [A3: list_a,B2: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( R2
= ( mult_l7073676228092353617t_unit @ R @ A3 @ B2 ) )
=> ( ( member_list_a @ A3 @ ( units_2932844235741507942t_unit @ R ) )
| ( member_list_a @ B2 @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ R @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleI
thf(fact_815_domain_Olagrange__aux__poly,axiom,
! [R: partia6043505979758434576t_unit,S2: set_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( finite_finite_set_a @ S2 )
=> ( ( ord_le3724670747650509150_set_a @ S2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( member_list_set_a @ ( lagran8431350174043022969t_unit @ R @ S2 ) @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) ) ) ) ) ).
% domain.lagrange_aux_poly
thf(fact_816_domain_Olagrange__aux__poly,axiom,
! [R: partia2670972154091845814t_unit,S2: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_list_a @ S2 )
=> ( ( ord_le8861187494160871172list_a @ S2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( lagran3534788790333317459t_unit @ R @ S2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ).
% domain.lagrange_aux_poly
thf(fact_817_domain_Olagrange__aux__poly,axiom,
! [R: partia2175431115845679010xt_a_b,S2: set_a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_a @ S2 )
=> ( ( ord_less_eq_set_a @ S2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( lagran9092808442999052491ux_a_b @ R @ S2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ).
% domain.lagrange_aux_poly
thf(fact_818_domain_Oprimeideal__iff__prime,axiom,
! [R: partia6043505979758434576t_unit,P2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ P2 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ( primei7645216761534224334t_unit @ ( cgenid6682780793756002467t_unit @ R @ P2 ) @ R )
= ( ring_r6795642478576035723t_unit @ R @ P2 ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_819_domain_Oprimeideal__iff__prime,axiom,
! [R: partia2670972154091845814t_unit,P2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P2 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( primei6309817859076077608t_unit @ ( cgenid9131348535277946915t_unit @ R @ P2 ) @ R )
= ( ring_r6430282645014804837t_unit @ R @ P2 ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_820_domain_Oprimeideal__iff__prime,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P2 @ ( 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 @ P2 ) @ R )
= ( ring_ring_prime_a_b @ R @ P2 ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_821_principal__domain_Odomain__iff__prime,axiom,
! [R: partia2175431115845679010xt_a_b,A: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ R @ ( cgenid547466209912283029xt_a_b @ R @ A ) ) )
= ( ring_ring_prime_a_b @ R @ A ) ) ) ) ).
% principal_domain.domain_iff_prime
thf(fact_822_principal__domain_Odomain__iff__prime,axiom,
! [R: partia2670972154091845814t_unit,A: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ A @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( domain1617769409708967785t_unit @ ( factRi3329376332477095402t_unit @ R @ ( cgenid9131348535277946915t_unit @ R @ A ) ) )
= ( ring_r6430282645014804837t_unit @ R @ A ) ) ) ) ).
% principal_domain.domain_iff_prime
thf(fact_823_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_824_ring__prime__def,axiom,
( ring_ring_prime_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,A6: a] :
( ( A6
!= ( zero_a_b @ R3 ) )
& ( prime_a_ring_ext_a_b @ R3 @ A6 ) ) ) ) ).
% ring_prime_def
thf(fact_825_ring__prime__def,axiom,
( ring_r6430282645014804837t_unit
= ( ^ [R3: partia2670972154091845814t_unit,A6: list_a] :
( ( A6
!= ( zero_l4142658623432671053t_unit @ R3 ) )
& ( prime_2011924034616061926t_unit @ R3 @ A6 ) ) ) ) ).
% ring_prime_def
thf(fact_826_principal__domain_Oprimeness__condition,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ P2 )
= ( ring_ring_prime_a_b @ R @ P2 ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_827_principal__domain_Oprimeness__condition,axiom,
! [R: partia2670972154091845814t_unit,P2: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ P2 )
= ( ring_r6430282645014804837t_unit @ R @ P2 ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_828_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ P2 )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ R @ P2 ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_829_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia2670972154091845814t_unit,P2: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ P2 )
=> ( maxima6585700282301356660t_unit @ ( cgenid9131348535277946915t_unit @ R @ P2 ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_830_x_Oring__primeI,axiom,
! [P2: list_a] :
( ( P2
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ).
% x.ring_primeI
thf(fact_831_x_Odimension_Ocases,axiom,
! [A1: nat,A22: set_list_a,A32: set_list_a] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ~ ! [V2: list_a,E4: set_list_a,N2: nat] :
( ( A1
= ( suc @ N2 ) )
=> ( ( A32
= ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A22 @ V2 @ E4 ) )
=> ( ( member_list_a @ V2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ~ ( member_list_a @ V2 @ E4 )
=> ~ ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N2 @ A22 @ E4 ) ) ) ) ) ) ) ).
% x.dimension.cases
thf(fact_832_x_Odimension_Osimps,axiom,
! [A1: nat,A22: set_list_a,A32: set_list_a] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A1 @ A22 @ A32 )
= ( ? [K5: set_list_a] :
( ( A1 = zero_zero_nat )
& ( A22 = K5 )
& ( A32
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
| ? [V4: list_a,E5: set_list_a,N4: nat,K5: set_list_a] :
( ( A1
= ( suc @ N4 ) )
& ( A22 = K5 )
& ( A32
= ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K5 @ V4 @ E5 ) )
& ( member_list_a @ V4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ~ ( member_list_a @ V4 @ E5 )
& ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N4 @ K5 @ E5 ) ) ) ) ).
% x.dimension.simps
thf(fact_833_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_834_ring__irreducibleE_I2_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( irredu6211895646901577903xt_a_b @ r @ R2 ) ) ) ).
% ring_irreducibleE(2)
thf(fact_835_zero__is__irreducible__iff__field,axiom,
( ( irredu6211895646901577903xt_a_b @ r @ ( zero_a_b @ r ) )
= ( field_a_b @ r ) ) ).
% zero_is_irreducible_iff_field
thf(fact_836_irreducible__prod__rI,axiom,
! [A: a,B: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_rI
thf(fact_837_irreducible__prod__lI,axiom,
! [B: a,A: a] :
( ( irredu6211895646901577903xt_a_b @ r @ B )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_lI
thf(fact_838_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_839_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_840_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_841_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_842_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_843_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_844_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_845_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_846_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_847_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_848_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_849_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_850_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_851_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_852_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_853_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_854_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_855_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_856_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_857_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_858_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_859_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_860_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_861_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_862_x_Ozero__dim,axiom,
! [K2: set_list_a] : ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ zero_zero_nat @ K2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ).
% x.zero_dim
thf(fact_863_x_Odimension__zero,axiom,
! [K2: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ zero_zero_nat @ K2 @ E )
=> ( E
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ).
% x.dimension_zero
thf(fact_864_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_865_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_866_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_867_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_868_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_869_add_Onat__pow__0,axiom,
! [X: a] :
( ( add_pow_a_b_nat @ r @ zero_zero_nat @ X )
= ( zero_a_b @ r ) ) ).
% add.nat_pow_0
thf(fact_870_x_Oadd_Onat__pow__0,axiom,
! [X: list_a] :
( ( add_po2640537187477214989it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ zero_zero_nat @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.add.nat_pow_0
thf(fact_871_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_872_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_873_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_874_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_875_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_876_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X2: nat,Y4: nat] :
( ( P @ X2 @ Y4 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_877_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_878_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_879_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_880_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_881_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_882_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_883_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_884_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_885_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_886_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_887_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_888_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_889_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_890_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_891_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_892_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_893_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_894_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_895_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_896_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_897_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_898_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_899_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_900_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_901_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_902_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_903_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_904_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_905_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_906_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_907_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_908_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_909_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_910_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_911_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_912_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_913_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_914_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_915_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_916_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_917_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_918_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_919_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_920_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_921_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_922_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_923_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_924_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_925_mult__mono_H,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_926_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_927_mult__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_928_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_929_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_930_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_931_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_932_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_933_euclidean__function,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ? [Q3: a,R4: a] :
( ( member_a @ Q3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ R4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( A
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ B @ Q3 ) @ R4 ) )
& ( ( R4
= ( zero_a_b @ r ) )
| ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ) ) ).
% euclidean_function
thf(fact_934_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_935_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_936_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_937_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_938_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_939_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_940_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_941_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_942_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_943_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_944_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_945_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_946_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_947_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_948_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_949_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_950_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_951_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_952_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_953_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_954_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_955_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_956_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_957_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_958_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_959_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_960_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_961_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_962_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_963_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_964_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_965_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_966_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_967_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_968_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_969_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_970_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_971_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_972_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_973_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_974_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_975_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_976_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_977_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_978_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_979_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_980_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_981_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_982_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_983_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_984_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_985_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_986_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_987_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_988_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_989_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_990_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_991_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_992_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_993_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_994_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_995_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_996_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_997_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_998_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_999_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_1000_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_1001_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1002_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_1003_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1004_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1005_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1006_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1007_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1008_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1009_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1010_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1011_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1012_euclidean__domainI,axiom,
! [Phi: a > nat] :
( ! [A3: a,B2: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ? [Q4: a,R5: a] :
( ( member_a @ Q4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ R5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( A3
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ B2 @ Q4 ) @ R5 ) )
& ( ( R5
= ( zero_a_b @ r ) )
| ( ord_less_nat @ ( Phi @ R5 ) @ ( Phi @ B2 ) ) ) ) ) )
=> ( ring_e8745995371659049232in_a_b @ r @ Phi ) ) ).
% euclidean_domainI
thf(fact_1013_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_1014_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1015_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1016_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1017_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1018_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1019_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1020_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1021_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1022_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1023_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1024_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1025_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1026_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1027_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1028_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1029_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1030_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1031_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1032_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1033_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1034_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1035_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1036_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_1037_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1038_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1039_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1040_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1041_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J2: nat,K4: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ K4 )
=> ( ( P @ I4 @ J2 )
=> ( ( P @ J2 @ K4 )
=> ( P @ I4 @ K4 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1042_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1043_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1044_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N4: nat] :
( ( ord_less_eq_nat @ M6 @ N4 )
& ( M6 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1045_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1046_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N4: nat] :
( ( ord_less_nat @ M6 @ N4 )
| ( M6 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1047_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1048_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1049_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1050_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1051_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1052_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1053_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1054_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1055_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1056_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1057_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1058_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1059_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1060_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1061_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1062_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_1063_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1064_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1065_less__not__refl3,axiom,
! [S: nat,T3: nat] :
( ( ord_less_nat @ S @ T3 )
=> ( S != T3 ) ) ).
% less_not_refl3
thf(fact_1066_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1067_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1068_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1069_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1070_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1071_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1072_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1073_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1074_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1075_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1076_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1077_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1078_add__strict__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_1079_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_1080_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1081_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1082_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1083_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1084_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1085_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1086_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_1087_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_1088_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_1089_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_1090_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_1091_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_1092_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1093_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1094_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1095_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1096_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_1097_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1098_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1099_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1100_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_1101_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1102_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1103_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1104_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1105_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1106_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1107_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1108_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1109_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_1110_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_1111_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_1112_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_1113_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_1114_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_1115_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_1116_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1117_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1118_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_1119_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_1120_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1121_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1122_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1123_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1124_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1125_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1126_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1127_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1128_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1129_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1130_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1131_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1132_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1133_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1134_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1135_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1136_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1137_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_1138_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1139_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1140_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1141_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1142_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1143_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1144_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1145_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1146_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1147_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1148_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1149_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1150_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1151_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1152_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1153_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1154_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1155_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1156_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1157_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1158_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1159_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1160_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1161_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K4: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1162_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M6: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1163_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1164_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1165_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1166_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1167_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1168_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1169_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1170_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1171_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1172_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1173_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1174_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1175_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1176_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1177_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1178_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1179_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1180_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1181_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K4 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1182_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1183_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1184_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1185_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1186_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1187_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1188_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1189_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_1190_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_1191_x_OboundD__carrier,axiom,
! [N: nat,F: nat > list_a,M: nat] :
( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_list_a @ ( F @ M ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.boundD_carrier
thf(fact_1192_alg__mult__gt__zero__iff__is__root,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) )
= ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_1193_boundD__carrier,axiom,
! [N: nat,F: nat > a,M: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_a @ ( F @ M ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_1194_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1195_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1196_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1197_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1198_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1199_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1200_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1201_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1202_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1203_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1204_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1205_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1206_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1207_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1208_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1209_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_1210_exists__irreducible__divisor,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ~ ! [B2: a] :
( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ B2 )
=> ~ ( factor8216151070175719842xt_a_b @ r @ B2 @ A ) ) ) ) ) ).
% exists_irreducible_divisor
thf(fact_1211_carrier__is__subfield,axiom,
subfield_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subfield
thf(fact_1212_univ__poly__is__principal,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% univ_poly_is_principal
thf(fact_1213_subring__props_I2_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K2 ) ) ).
% subring_props(2)
thf(fact_1214_subring__props_I7_J,axiom,
! [K2: set_a,H12: a,H22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H12 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( add_a_b @ r @ H12 @ H22 ) @ K2 ) ) ) ) ).
% subring_props(7)
thf(fact_1215_subring__props_I6_J,axiom,
! [K2: set_a,H12: a,H22: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ H12 @ K2 )
=> ( ( member_a @ H22 @ K2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H22 ) @ K2 ) ) ) ) ).
% subring_props(6)
thf(fact_1216_subring__props_I4_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( K2 != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_1217_subring__props_I3_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K2 ) ) ).
% subring_props(3)
thf(fact_1218_subring__props_I1_J,axiom,
! [K2: set_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ord_less_eq_set_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_1219_divides__trans,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( factor8216151070175719842xt_a_b @ r @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ C ) ) ) ) ).
% divides_trans
thf(fact_1220_zero__divides,axiom,
! [A: a] :
( ( factor8216151070175719842xt_a_b @ r @ ( zero_a_b @ r ) @ A )
= ( A
= ( zero_a_b @ r ) ) ) ).
% zero_divides
thf(fact_1221_divides__zero,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( zero_a_b @ r ) ) ) ).
% divides_zero
thf(fact_1222_local_Odivides__mult,axiom,
! [A: a,C: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% local.divides_mult
thf(fact_1223_divides__prod__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ) ).
% divides_prod_l
thf(fact_1224_divides__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ).
% divides_prod_r
thf(fact_1225_one__divides,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ A ) ) ).
% one_divides
thf(fact_1226_divides__unit,axiom,
! [A: a,U: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ U )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% divides_unit
thf(fact_1227_unit__divides,axiom,
! [U: a,A: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ U @ A ) ) ) ).
% unit_divides
thf(fact_1228_pprime__iff__pirreducible,axiom,
! [K2: set_a,P2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P2 )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P2 ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_1229_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_1230_divides__one,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ ( one_a_ring_ext_a_b @ r ) )
= ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% divides_one
thf(fact_1231_to__contain__is__to__divide,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( cgenid547466209912283029xt_a_b @ r @ B ) @ ( cgenid547466209912283029xt_a_b @ r @ A ) )
= ( factor8216151070175719842xt_a_b @ r @ A @ B ) ) ) ) ).
% to_contain_is_to_divide
thf(fact_1232_pprimeE_I2_J,axiom,
! [K2: set_a,P2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P2 )
=> ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ) ) ) ).
% pprimeE(2)
thf(fact_1233_subfield__m__inv__simprule,axiom,
! [K2: set_a,K: a,A: a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_a @ K @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K @ A ) @ K2 )
=> ( member_a @ A @ K2 ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_1234_mult__divides,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ B ) ) ) ) ) ).
% mult_divides
thf(fact_1235_divides__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ A ) ) ).
% divides_refl
thf(fact_1236_divides__mult__lI,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% divides_mult_lI
thf(fact_1237_divides__mult__rI,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% divides_mult_rI
thf(fact_1238_isgcd__divides__r,axiom,
! [B: a,A: a] :
( ( factor8216151070175719842xt_a_b @ r @ B @ A )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ B @ A @ B ) ) ) ) ).
% isgcd_divides_r
thf(fact_1239_isgcd__divides__l,axiom,
! [A: a,B: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ A @ A @ B ) ) ) ) ).
% isgcd_divides_l
thf(fact_1240_x_Odivides__trans,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) ) ) ) ).
% x.divides_trans
thf(fact_1241_x_Ozero__divides,axiom,
! [A: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.zero_divides
thf(fact_1242_x_Odivides__zero,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.divides_zero
thf(fact_1243_x_Odivides__prod__r,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) ) ) ) ) ).
% x.divides_prod_r
thf(fact_1244_x_Odivides__prod__l,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ) ).
% x.divides_prod_l
thf(fact_1245_x_Odivides__mult,axiom,
! [A: list_a,C: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ).
% x.divides_mult
thf(fact_1246_x_Ounit__divides,axiom,
! [U: list_a,A: list_a] :
( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ A ) ) ) ).
% x.unit_divides
thf(fact_1247_x_Odivides__unit,axiom,
! [A: list_a,U: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ U )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.divides_unit
thf(fact_1248_x_Oone__divides,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A ) ) ).
% x.one_divides
thf(fact_1249_x_Odivides__one,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.divides_one
thf(fact_1250_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_1251_x_Oto__contain__is__to__divide,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ).
% x.to_contain_is_to_divide
thf(fact_1252_x_Odivides__refl,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ A ) ) ).
% x.divides_refl
thf(fact_1253_x_Odivides__mult__rI,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) ) ) ) ) ) ).
% x.divides_mult_rI
thf(fact_1254_x_Odivides__mult__lI,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ).
% x.divides_mult_lI
thf(fact_1255_x_Oisgcd__divides__r,axiom,
! [B: list_a,A: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( isgcd_1118609098697428327t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A @ B ) ) ) ) ).
% x.isgcd_divides_r
thf(fact_1256_x_Oisgcd__divides__l,axiom,
! [A: list_a,B: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( isgcd_1118609098697428327t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ A @ B ) ) ) ) ).
% x.isgcd_divides_l
thf(fact_1257_rupture__is__field__iff__pirreducible,axiom,
! [K2: set_a,P2: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ r @ K2 @ P2 ) )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K2 ) @ P2 ) ) ) ) ).
% rupture_is_field_iff_pirreducible
thf(fact_1258_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_1259_long__division__add_I1_J,axiom,
! [K2: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( polynomial_pdiv_a_b @ r @ A @ Q ) @ ( polynomial_pdiv_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(1)
thf(fact_1260_long__division__closed_I1_J,axiom,
! [K2: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_1261_pdiv__pmod,axiom,
! [K2: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K2 @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K2 ) @ Q @ ( polynomial_pdiv_a_b @ r @ P2 @ Q ) ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q ) ) ) ) ) ) ).
% pdiv_pmod
thf(fact_1262_divides__imp__divides__mult,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ).
% divides_imp_divides_mult
thf(fact_1263_mult__of_OUnits__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.Units_closed
thf(fact_1264_mult__of_Ogcdof__exists,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ C2 @ A @ B ) ) ) ) ).
% mult_of.gcdof_exists
thf(fact_1265_mult__of_Omonoid__cancel__axioms,axiom,
monoid1999574367301118026t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.monoid_cancel_axioms
thf(fact_1266_zero__is__prime_I2_J,axiom,
prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ ( zero_a_b @ r ) ).
% zero_is_prime(2)
thf(fact_1267_mult__of_Oprod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.prod_unit_l
thf(fact_1268_mult__of_Oprod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.prod_unit_r
thf(fact_1269_mult__of_Ounit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.unit_factor
thf(fact_1270_mult__of_Omonoid__cancelI,axiom,
( ! [A3: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A3 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B2 ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A3 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B2 @ C2 ) )
=> ( ( member_a @ A3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid1999574367301118026t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.monoid_cancelI
thf(fact_1271_mult__of_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ C )
= ( mult_a_ring_ext_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A = B ) ) ) ) ) ).
% mult_of.r_cancel
thf(fact_1272_mult__of_Om__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ 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 ) ) ) ) ) ) ).
% mult_of.m_lcomm
% Conjectures (1)
thf(conj_0,conjecture,
member_list_a @ p @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
%------------------------------------------------------------------------------