TPTP Problem File: SLH0298^1.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 : Universal_Hash_Families/0028_Field/prob_00228_008161__18344400_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1437 ( 357 unt; 160 typ; 0 def)
% Number of atoms : 4353 (1299 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 17273 ( 141 ~; 26 |; 125 &;14446 @)
% ( 0 <=>;2535 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 8 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 373 ( 373 >; 0 *; 0 +; 0 <<)
% Number of symbols : 148 ( 147 usr; 9 con; 0-4 aty)
% Number of variables : 3474 ( 46 ^;3355 !; 73 ?;3474 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:42:35.077
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
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thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Nat__Onat_Mt__Group__Omonoid__Omonoid____ext_It__Nat__Onat_Mt__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
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% Explicit typings (147)
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thf(sy_c_Field_Omod__ring,type,
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thf(sy_c_Field_Ozfact__iso__inv,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
bot_bot_set_set_int: set_set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
ord_le5995675665013768039et_int: set_nat_set_int > set_nat_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_QuotRing_Oring__iso_001t__Nat__Onat_001t__Product____Type__Ounit_001t__Nat__Onat_001t__Product____Type__Ounit,type,
ring_i3628375019190340560t_unit: partia4692342223508353374t_unit > partia4692342223508353374t_unit > set_nat_nat ).
thf(sy_c_QuotRing_Oring__iso_001t__Nat__Onat_001t__Product____Type__Ounit_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_i1863809825068120638t_unit: partia4692342223508353374t_unit > partia4934656038542163276t_unit > set_nat_set_int ).
thf(sy_c_QuotRing_Oring__iso_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Nat__Onat_001t__Product____Type__Ounit,type,
ring_i6162119212153773794t_unit: partia4934656038542163276t_unit > partia4692342223508353374t_unit > set_set_int_nat ).
thf(sy_c_QuotRing_Oring__iso_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_i7857697894474155344t_unit: partia4934656038542163276t_unit > partia4934656038542163276t_unit > set_set_int_set_int ).
thf(sy_c_Ring_Oa__inv_001t__Nat__Onat_001t__Product____Type__Ounit,type,
a_inv_2472168910397739247t_unit: partia4692342223508353374t_unit > nat > nat ).
thf(sy_c_Ring_Oa__inv_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
a_inv_5951419416477254493t_unit: partia4934656038542163276t_unit > set_int > set_int ).
thf(sy_c_Ring_Oa__minus_001t__Nat__Onat_001t__Product____Type__Ounit,type,
a_minu1751788497103602224t_unit: partia4692342223508353374t_unit > nat > nat > nat ).
thf(sy_c_Ring_Oa__minus_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
a_minu5974516859897376926t_unit: partia4934656038542163276t_unit > set_int > set_int > set_int ).
thf(sy_c_Ring_Oabelian__group_001t__Nat__Onat_001t__Product____Type__Ounit,type,
abelia406319425121669262t_unit: partia4692342223508353374t_unit > $o ).
thf(sy_c_Ring_Oabelian__group_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
abelia23968383328945916t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Oabelian__monoid_001t__Nat__Onat_001t__Product____Type__Ounit,type,
abelia362511065248671243t_unit: partia4692342223508353374t_unit > $o ).
thf(sy_c_Ring_Oabelian__monoid_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
abelia3815030880812984441t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Oadd__pow_001t__Nat__Onat_001t__Product____Type__Ounit_001t__Int__Oint,type,
add_po2420080144553951285it_int: partia4692342223508353374t_unit > int > nat > nat ).
thf(sy_c_Ring_Oadd__pow_001t__Nat__Onat_001t__Product____Type__Ounit_001t__Nat__Onat,type,
add_po2422570615063001561it_nat: partia4692342223508353374t_unit > nat > nat > nat ).
thf(sy_c_Ring_Oadd__pow_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Int__Oint,type,
add_po7581009264371422883it_int: partia4934656038542163276t_unit > int > set_int > set_int ).
thf(sy_c_Ring_Oadd__pow_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Nat__Onat,type,
add_po7583499734880473159it_nat: partia4934656038542163276t_unit > nat > set_int > set_int ).
thf(sy_c_Ring_Oring_001t__Nat__Onat_001t__Product____Type__Ounit,type,
ring_n9194430563101542159t_unit: partia4692342223508353374t_unit > $o ).
thf(sy_c_Ring_Oring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_s5316885176909347197t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Oring_Oadd_001t__Nat__Onat_001t__Product____Type__Ounit,type,
add_nat_Product_unit: partia4692342223508353374t_unit > nat > nat > nat ).
thf(sy_c_Ring_Oring_Oadd_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
add_se5859248395121729892t_unit: partia4934656038542163276t_unit > set_int > set_int > set_int ).
thf(sy_c_Ring_Oring_Ozero_001t__Nat__Onat_001t__Product____Type__Ounit,type,
zero_n5149899317435570679t_unit: partia4692342223508353374t_unit > nat ).
thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
zero_s6269048424454532197t_unit: partia4934656038542163276t_unit > set_int ).
thf(sy_c_Ring_Oring__hom_001t__Nat__Onat_001t__Product____Type__Ounit_001t__Nat__Onat_001t__Product____Type__Ounit,type,
ring_h4412161176302437050t_unit: partia4692342223508353374t_unit > partia4692342223508353374t_unit > set_nat_nat ).
thf(sy_c_Ring_Oring__hom_001t__Nat__Onat_001t__Product____Type__Ounit_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_h4752909569380436264t_unit: partia4692342223508353374t_unit > partia4934656038542163276t_unit > set_nat_set_int ).
thf(sy_c_Ring_Oring__hom_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Nat__Onat_001t__Product____Type__Ounit,type,
ring_h9051218956466089420t_unit: partia4934656038542163276t_unit > partia4692342223508353374t_unit > set_set_int_nat ).
thf(sy_c_Ring_Oring__hom_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_h3404898052528352314t_unit: partia4934656038542163276t_unit > partia4934656038542163276t_unit > set_set_int_set_int ).
thf(sy_c_Ring_Osemiring_001t__Nat__Onat_001t__Product____Type__Ounit,type,
semiri3921172975686117281t_unit: partia4692342223508353374t_unit > $o ).
thf(sy_c_Ring_Osemiring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
semiri8708897239777792527t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring__Characteristic_Ochar_001t__Nat__Onat_001t__Product____Type__Ounit,type,
ring_c2111903209400434062t_unit: partia4692342223508353374t_unit > nat ).
thf(sy_c_Ring__Characteristic_Oint__embed_001t__Nat__Onat_001t__Product____Type__Ounit,type,
ring_i4887507503946925882t_unit: partia4692342223508353374t_unit > int > nat ).
thf(sy_c_Ring__Characteristic_Ozfact__iso,type,
ring_zfact_iso: nat > nat > set_int ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
collect_nat_set_int: ( ( nat > set_int ) > $o ) > set_nat_set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
insert_nat_set_int: ( nat > set_int ) > set_nat_set_int > set_nat_set_int ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Int__Oint_J,type,
insert_set_int: set_int > set_set_int > set_set_int ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Subrings_Osubcring_001t__Nat__Onat_001t__Product____Type__Ounit,type,
subcri1627753237249443161t_unit: set_nat > partia4692342223508353374t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
subcri1024317279029940167t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001t__Nat__Onat_001t__Product____Type__Ounit,type,
subdom2148668005855505734t_unit: set_nat > partia4692342223508353374t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
subdom1520866149873910708t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001t__Nat__Onat_001t__Product____Type__Ounit,type,
subfie4892355163478727762t_unit: set_nat > partia4692342223508353374t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
subfie3888952257595785920t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Subrings_Osubring_001t__Nat__Onat_001t__Product____Type__Ounit,type,
subrin2893992908230074586t_unit: set_nat > partia4692342223508353374t_unit > $o ).
thf(sy_c_Subrings_Osubring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
subrin7689096310803670856t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
member_nat_set_int: ( nat > set_int ) > set_nat_set_int > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J,type,
member_set_int_nat: ( set_int > nat ) > set_set_int_nat > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
member5205197933313416826et_int: ( set_int > set_int ) > set_set_int_set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1276)
thf(fact_0_s_Oring__iso__restrict,axiom,
! [F: nat > set_int,S: partia4934656038542163276t_unit,G: nat > set_int] :
( ( member_nat_set_int @ F @ ( ring_i1863809825068120638t_unit @ ( mod_ring @ n ) @ S ) )
=> ( ! [R: nat] :
( ( member_nat @ R @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( F @ R )
= ( G @ R ) ) )
=> ( member_nat_set_int @ G @ ( ring_i1863809825068120638t_unit @ ( mod_ring @ n ) @ S ) ) ) ) ).
% s.ring_iso_restrict
thf(fact_1__092_060open_062inv__into_A_Icarrier_A_IZFact_A_Iint_An_J_J_J_A_Izfact__iso__inv_An_J_A_092_060in_062_Aring__iso_A_Imod__ring_An_J_A_IZFact_A_Iint_An_J_J_092_060close_062,axiom,
member_nat_set_int @ ( hilber5958887739343024896nt_nat @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zfact_iso_inv @ n ) ) @ ( ring_i1863809825068120638t_unit @ ( mod_ring @ n ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% \<open>inv_into (carrier (ZFact (int n))) (zfact_iso_inv n) \<in> ring_iso (mod_ring n) (ZFact (int n))\<close>
thf(fact_2__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Acarrier_A_Imod__ring_An_J_A_092_060Longrightarrow_062_Ainv__into_A_Icarrier_A_IZFact_A_Iint_An_J_J_J_A_Izfact__iso__inv_An_J_Ax_A_061_Azfact__iso_An_Ax_092_060close_062,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( hilber5958887739343024896nt_nat @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zfact_iso_inv @ n ) @ X )
= ( ring_zfact_iso @ n @ X ) ) ) ).
% \<open>\<And>x. x \<in> carrier (mod_ring n) \<Longrightarrow> inv_into (carrier (ZFact (int n))) (zfact_iso_inv n) x = zfact_iso n x\<close>
thf(fact_3_s_Osemiring__axioms,axiom,
semiri3921172975686117281t_unit @ ( mod_ring @ n ) ).
% s.semiring_axioms
thf(fact_4_r,axiom,
ring_s5316885176909347197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r
thf(fact_5_s_Oonepideal,axiom,
princi4652470909602072491t_unit @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) @ ( mod_ring @ n ) ).
% s.onepideal
thf(fact_6_s_Oabelian__monoid__axioms,axiom,
abelia362511065248671243t_unit @ ( mod_ring @ n ) ).
% s.abelian_monoid_axioms
thf(fact_7_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_8_s_Oring__axioms,axiom,
ring_n9194430563101542159t_unit @ ( mod_ring @ n ) ).
% s.ring_axioms
thf(fact_9_s_Ois__abelian__group,axiom,
abelia406319425121669262t_unit @ ( mod_ring @ n ) ).
% s.is_abelian_group
thf(fact_10_s_Ocgenideal__self,axiom,
! [I: nat] :
( ( member_nat @ I @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ I @ ( cgenid8693976350862946099t_unit @ ( mod_ring @ n ) @ I ) ) ) ).
% s.cgenideal_self
thf(fact_11_s_Oassociated__sym,axiom,
! [A: nat,B: nat] :
( ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ A @ B )
=> ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ B @ A ) ) ).
% s.associated_sym
thf(fact_12_s_Omonoid__axioms,axiom,
monoid4477821099152670945t_unit @ ( mod_ring @ n ) ).
% s.monoid_axioms
thf(fact_13_s_Ocarrier__not__empty,axiom,
( ( partia3499330772048238685t_unit @ ( mod_ring @ n ) )
!= bot_bot_set_nat ) ).
% s.carrier_not_empty
thf(fact_14_s_Oassociated__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ A @ B )
=> ( ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ B @ C )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ A @ C ) ) ) ) ) ).
% s.associated_trans
thf(fact_15_s_Oassoc__subst,axiom,
! [A: nat,B: nat,F: nat > nat] :
( ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ A @ B )
=> ( ! [A2: nat,B2: nat] :
( ( ( member_nat @ A2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
& ( member_nat @ B2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
& ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ A2 @ B2 ) )
=> ( ( member_nat @ ( F @ A2 ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
& ( member_nat @ ( F @ B2 ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
& ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% s.assoc_subst
thf(fact_16_s_Oassociated__refl,axiom,
! [A: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ A @ A ) ) ).
% s.associated_refl
thf(fact_17_ring__iso__set__sym,axiom,
! [R2: partia4934656038542163276t_unit,H: set_int > nat,S: partia4692342223508353374t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ R2 @ S ) )
=> ( member_nat_set_int @ ( hilber5958887739343024896nt_nat @ ( partia966996272515721803t_unit @ R2 ) @ H ) @ ( ring_i1863809825068120638t_unit @ S @ R2 ) ) ) ) ).
% ring_iso_set_sym
thf(fact_18_ring__iso__set__sym,axiom,
! [R2: partia4692342223508353374t_unit,H: nat > set_int,S: partia4934656038542163276t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R2 @ S ) )
=> ( member_set_int_nat @ ( hilber3159750049796175616et_int @ ( partia3499330772048238685t_unit @ R2 ) @ H ) @ ( ring_i6162119212153773794t_unit @ S @ R2 ) ) ) ) ).
% ring_iso_set_sym
thf(fact_19_s_Oproperfactor__cong__r,axiom,
! [X: nat,Y: nat,Y2: nat] :
( ( proper2699154196232879280t_unit @ ( mod_ring @ n ) @ X @ Y )
=> ( ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ Y @ Y2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( proper2699154196232879280t_unit @ ( mod_ring @ n ) @ X @ Y2 ) ) ) ) ) ) ).
% s.properfactor_cong_r
thf(fact_20_s_Oproperfactor__cong__l,axiom,
! [X2: nat,X: nat,Y: nat] :
( ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ X2 @ X )
=> ( ( proper2699154196232879280t_unit @ ( mod_ring @ n ) @ X @ Y )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( proper2699154196232879280t_unit @ ( mod_ring @ n ) @ X2 @ Y ) ) ) ) ) ) ).
% s.properfactor_cong_l
thf(fact_21_s_Omult__cong__r,axiom,
! [B: nat,B3: nat,A: nat] :
( ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ B @ B3 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B3 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ B ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ B3 ) ) ) ) ) ) ).
% s.mult_cong_r
thf(fact_22_ring_Oonepideal,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( princi8860937869964495385t_unit @ ( partia966996272515721803t_unit @ R2 ) @ R2 ) ) ).
% ring.onepideal
thf(fact_23_ring_Oonepideal,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( princi4652470909602072491t_unit @ ( partia3499330772048238685t_unit @ R2 ) @ R2 ) ) ).
% ring.onepideal
thf(fact_24_ring_Oring__iso__restrict,axiom,
! [R2: partia4692342223508353374t_unit,F: nat > set_int,S: partia4934656038542163276t_unit,G: nat > set_int] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat_set_int @ F @ ( ring_i1863809825068120638t_unit @ R2 @ S ) )
=> ( ! [R: nat] :
( ( member_nat @ R @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( F @ R )
= ( G @ R ) ) )
=> ( member_nat_set_int @ G @ ( ring_i1863809825068120638t_unit @ R2 @ S ) ) ) ) ) ).
% ring.ring_iso_restrict
thf(fact_25_ring_Ocgenideal__self,axiom,
! [R2: partia4934656038542163276t_unit,I: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ I @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ I @ ( cgenid8502489213727343375t_unit @ R2 @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_26_ring_Ocgenideal__self,axiom,
! [R2: partia4692342223508353374t_unit,I: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ I @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ I @ ( cgenid8693976350862946099t_unit @ R2 @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_27_monoid_Oassoc__subst,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int,F: set_int > set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( associ3816808824722140549t_unit @ G2 @ A @ B )
=> ( ! [A2: set_int,B2: set_int] :
( ( ( member_set_int @ A2 @ ( partia966996272515721803t_unit @ G2 ) )
& ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ G2 ) )
& ( associ3816808824722140549t_unit @ G2 @ A2 @ B2 ) )
=> ( ( member_set_int @ ( F @ A2 ) @ ( partia966996272515721803t_unit @ G2 ) )
& ( member_set_int @ ( F @ B2 ) @ ( partia966996272515721803t_unit @ G2 ) )
& ( associ3816808824722140549t_unit @ G2 @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( associ3816808824722140549t_unit @ G2 @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ) ).
% monoid.assoc_subst
thf(fact_28_monoid_Oassoc__subst,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat,F: nat > nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( associ4357985397585971625t_unit @ G2 @ A @ B )
=> ( ! [A2: nat,B2: nat] :
( ( ( member_nat @ A2 @ ( partia3499330772048238685t_unit @ G2 ) )
& ( member_nat @ B2 @ ( partia3499330772048238685t_unit @ G2 ) )
& ( associ4357985397585971625t_unit @ G2 @ A2 @ B2 ) )
=> ( ( member_nat @ ( F @ A2 ) @ ( partia3499330772048238685t_unit @ G2 ) )
& ( member_nat @ ( F @ B2 ) @ ( partia3499330772048238685t_unit @ G2 ) )
& ( associ4357985397585971625t_unit @ G2 @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( associ4357985397585971625t_unit @ G2 @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ) ).
% monoid.assoc_subst
thf(fact_29_monoid_Oassociated__refl,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( associ3816808824722140549t_unit @ G2 @ A @ A ) ) ) ).
% monoid.associated_refl
thf(fact_30_monoid_Oassociated__refl,axiom,
! [G2: partia4692342223508353374t_unit,A: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( associ4357985397585971625t_unit @ G2 @ A @ A ) ) ) ).
% monoid.associated_refl
thf(fact_31_monoid_Oassociated__trans,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int,C: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( associ3816808824722140549t_unit @ G2 @ A @ B )
=> ( ( associ3816808824722140549t_unit @ G2 @ B @ C )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G2 ) )
=> ( associ3816808824722140549t_unit @ G2 @ A @ C ) ) ) ) ) ) ).
% monoid.associated_trans
thf(fact_32_monoid_Oassociated__trans,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat,C: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( associ4357985397585971625t_unit @ G2 @ A @ B )
=> ( ( associ4357985397585971625t_unit @ G2 @ B @ C )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( associ4357985397585971625t_unit @ G2 @ A @ C ) ) ) ) ) ) ).
% monoid.associated_trans
thf(fact_33_s_Om__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y ) @ Z )
= ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Y @ Z ) ) ) ) ) ) ).
% s.m_assoc
thf(fact_34_s_Oproperfactor__prod__r,axiom,
! [A: nat,B: nat,C: nat] :
( ( proper2699154196232879280t_unit @ ( mod_ring @ n ) @ A @ B )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( proper2699154196232879280t_unit @ ( mod_ring @ n ) @ A @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ B @ C ) ) ) ) ) ) ).
% s.properfactor_prod_r
thf(fact_35_s_Om__closed,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.m_closed
thf(fact_36_monoid_Oproperfactor__prod__r,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int,C: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( proper1002977052347345036t_unit @ G2 @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G2 ) )
=> ( proper1002977052347345036t_unit @ G2 @ A @ ( mult_s3864001451298473021t_unit @ G2 @ B @ C ) ) ) ) ) ) ) ).
% monoid.properfactor_prod_r
thf(fact_37_monoid_Oproperfactor__prod__r,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat,C: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( proper2699154196232879280t_unit @ G2 @ A @ B )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( proper2699154196232879280t_unit @ G2 @ A @ ( mult_n6028127365542633569t_unit @ G2 @ B @ C ) ) ) ) ) ) ) ).
% monoid.properfactor_prod_r
thf(fact_38_ring__iso__memE_I2_J,axiom,
! [H: set_int > nat,R2: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,X: set_int,Y: set_int] :
( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ R2 @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( H @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Y ) )
= ( mult_n6028127365542633569t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_39_ring__iso__memE_I2_J,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( member_nat_nat @ H @ ( ring_i3628375019190340560t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) )
= ( mult_n6028127365542633569t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_40_ring__iso__memE_I2_J,axiom,
! [H: nat > set_int,R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,X: nat,Y: nat] :
( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) )
= ( mult_s3864001451298473021t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_41_monoid_Omult__cong__r,axiom,
! [G2: partia4934656038542163276t_unit,B: set_int,B3: set_int,A: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( associ3816808824722140549t_unit @ G2 @ B @ B3 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B3 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( associ3816808824722140549t_unit @ G2 @ ( mult_s3864001451298473021t_unit @ G2 @ A @ B ) @ ( mult_s3864001451298473021t_unit @ G2 @ A @ B3 ) ) ) ) ) ) ) ).
% monoid.mult_cong_r
thf(fact_42_monoid_Omult__cong__r,axiom,
! [G2: partia4692342223508353374t_unit,B: nat,B3: nat,A: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( associ4357985397585971625t_unit @ G2 @ B @ B3 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B3 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( associ4357985397585971625t_unit @ G2 @ ( mult_n6028127365542633569t_unit @ G2 @ A @ B ) @ ( mult_n6028127365542633569t_unit @ G2 @ A @ B3 ) ) ) ) ) ) ) ).
% monoid.mult_cong_r
thf(fact_43_monoid_Oproperfactor__cong__r,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Y2: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( proper1002977052347345036t_unit @ G2 @ X @ Y )
=> ( ( associ3816808824722140549t_unit @ G2 @ Y @ Y2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( proper1002977052347345036t_unit @ G2 @ X @ Y2 ) ) ) ) ) ) ) ).
% monoid.properfactor_cong_r
thf(fact_44_monoid_Oproperfactor__cong__r,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat,Y2: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( proper2699154196232879280t_unit @ G2 @ X @ Y )
=> ( ( associ4357985397585971625t_unit @ G2 @ Y @ Y2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( proper2699154196232879280t_unit @ G2 @ X @ Y2 ) ) ) ) ) ) ) ).
% monoid.properfactor_cong_r
thf(fact_45_monoid_Oproperfactor__cong__l,axiom,
! [G2: partia4934656038542163276t_unit,X2: set_int,X: set_int,Y: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( associ3816808824722140549t_unit @ G2 @ X2 @ X )
=> ( ( proper1002977052347345036t_unit @ G2 @ X @ Y )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( proper1002977052347345036t_unit @ G2 @ X2 @ Y ) ) ) ) ) ) ) ).
% monoid.properfactor_cong_l
thf(fact_46_monoid_Oproperfactor__cong__l,axiom,
! [G2: partia4692342223508353374t_unit,X2: nat,X: nat,Y: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( associ4357985397585971625t_unit @ G2 @ X2 @ X )
=> ( ( proper2699154196232879280t_unit @ G2 @ X @ Y )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( proper2699154196232879280t_unit @ G2 @ X2 @ Y ) ) ) ) ) ) ) ).
% monoid.properfactor_cong_l
thf(fact_47_principalideal_Ois__principalideal,axiom,
! [I2: set_nat,R2: partia4692342223508353374t_unit] :
( ( princi4652470909602072491t_unit @ I2 @ R2 )
=> ( princi4652470909602072491t_unit @ I2 @ R2 ) ) ).
% principalideal.is_principalideal
thf(fact_48_monoid_Oassociated__sym,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( associ4357985397585971625t_unit @ G2 @ A @ B )
=> ( associ4357985397585971625t_unit @ G2 @ B @ A ) ) ) ).
% monoid.associated_sym
thf(fact_49_ring__iso__memE_I1_J,axiom,
! [H: set_int > set_int,R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,X: set_int] :
( ( member5205197933313416826et_int @ H @ ( ring_i7857697894474155344t_unit @ R2 @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( H @ X ) @ ( partia966996272515721803t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_50_ring__iso__memE_I1_J,axiom,
! [H: set_int > nat,R2: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,X: set_int] :
( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ R2 @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_nat @ ( H @ X ) @ ( partia3499330772048238685t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_51_ring__iso__memE_I1_J,axiom,
! [H: nat > set_int,R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,X: nat] :
( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_set_int @ ( H @ X ) @ ( partia966996272515721803t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_52_ring__iso__memE_I1_J,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit,X: nat] :
( ( member_nat_nat @ H @ ( ring_i3628375019190340560t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( H @ X ) @ ( partia3499330772048238685t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_53_s_Omonoid__cancelI,axiom,
( ! [A2: nat,B2: nat,C2: nat] :
( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ C2 @ A2 )
= ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ C2 @ B2 ) )
=> ( ( member_nat @ A2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: nat,B2: nat,C2: nat] :
( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A2 @ C2 )
= ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ B2 @ C2 ) )
=> ( ( member_nat @ A2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid2713784563552164055t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.monoid_cancelI
thf(fact_54_s_Oadd__pow__ldistr__int,axiom,
! [A: nat,B: nat,K: int] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ K @ A ) @ B )
= ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ K @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ B ) ) ) ) ) ).
% s.add_pow_ldistr_int
thf(fact_55_s_Oadd__pow__rdistr__int,axiom,
! [A: nat,B: nat,K: int] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ K @ B ) )
= ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ K @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ B ) ) ) ) ) ).
% s.add_pow_rdistr_int
thf(fact_56_s_OassociatedI2,axiom,
! [U: nat,A: nat,B: nat] :
( ( member_nat @ U @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( A
= ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ B @ U ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ A @ B ) ) ) ) ).
% s.associatedI2
thf(fact_57_mem__Collect__eq,axiom,
! [A: nat > set_int,P: ( nat > set_int ) > $o] :
( ( member_nat_set_int @ A @ ( collect_nat_set_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_58_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_59_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A3: set_nat_set_int] :
( ( collect_nat_set_int
@ ^ [X3: nat > set_int] : ( member_nat_set_int @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A3: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_63_s_OassociatedI2_H,axiom,
! [A: nat,B: nat,U: nat] :
( ( A
= ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ B @ U ) )
=> ( ( member_nat @ U @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ A @ B ) ) ) ) ).
% s.associatedI2'
thf(fact_64_s_Ol__distr,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Z ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Y @ Z ) ) ) ) ) ) ).
% s.l_distr
thf(fact_65_s_Or__distr,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Z @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Z @ X ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Z @ Y ) ) ) ) ) ) ).
% s.r_distr
thf(fact_66_s_Oadd__pow__ldistr,axiom,
! [A: nat,B: nat,K: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ K @ A ) @ B )
= ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ K @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ B ) ) ) ) ) ).
% s.add_pow_ldistr
thf(fact_67_s_Oadd__pow__rdistr,axiom,
! [A: nat,B: nat,K: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ K @ B ) )
= ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ K @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ B ) ) ) ) ) ).
% s.add_pow_rdistr
thf(fact_68_s_Oinv__unique,axiom,
! [Y: nat,X: nat,Y2: nat] :
( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Y @ X )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y2 )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% s.inv_unique
thf(fact_69_s_Oone__unique,axiom,
! [U: nat] :
( ( member_nat @ U @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ U @ X4 )
= X4 ) )
=> ( U
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.one_unique
thf(fact_70_s_Oadd_Or__cancel,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ A @ C )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ B @ C ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( A = B ) ) ) ) ) ).
% s.add.r_cancel
thf(fact_71_s_Oadd_Om__lcomm,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ Z ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Z ) ) ) ) ) ) ).
% s.add.m_lcomm
thf(fact_72_s_Oadd_Om__comm,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ X ) ) ) ) ).
% s.add.m_comm
thf(fact_73_s_Oadd_Om__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ Z ) ) ) ) ) ) ).
% s.add.m_assoc
thf(fact_74_s_Oadd_Ol__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ C @ A )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ C @ B ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( A = B ) ) ) ) ) ).
% s.add.l_cancel
thf(fact_75_s_OUnits__closed,axiom,
! [X: nat] :
( ( member_nat @ X @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.Units_closed
thf(fact_76_s_OUnits__assoc,axiom,
! [A: nat,B: nat] :
( ( member_nat @ A @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( associ4357985397585971625t_unit @ ( mod_ring @ n ) @ A @ B ) ) ) ).
% s.Units_assoc
thf(fact_77_s_Oadd_Opow__mult__distrib,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ X ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ Y ) ) ) ) ) ) ).
% s.add.pow_mult_distrib
thf(fact_78_s_Oadd_Onat__pow__distrib,axiom,
! [X: nat,Y: nat,N: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ Y ) ) ) ) ) ).
% s.add.nat_pow_distrib
thf(fact_79_s_Oadd_Onat__pow__comm,axiom,
! [X: nat,N: nat,M: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ M @ X ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ M @ X ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) ) ) ) ).
% s.add.nat_pow_comm
thf(fact_80_s_Oadd_Ogroup__commutes__pow,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ X ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) @ Y )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) ) ) ) ) ) ).
% s.add.group_commutes_pow
thf(fact_81_s_Oadd_Oint__pow__mult__distrib,axiom,
! [X: nat,Y: nat,I: int] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ X ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ X ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ Y ) ) ) ) ) ) ).
% s.add.int_pow_mult_distrib
thf(fact_82_s_Oadd_Oint__pow__distrib,axiom,
! [X: nat,Y: nat,I: int] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ X ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ Y ) ) ) ) ) ).
% s.add.int_pow_distrib
thf(fact_83_s_Oprod__unit__r,axiom,
! [A: nat,B: nat] :
( ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ B ) @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ A @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) ) ) ) ) ) ).
% s.prod_unit_r
thf(fact_84_s_Oprod__unit__l,axiom,
! [A: nat,B: nat] :
( ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ B ) @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ A @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ B @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) ) ) ) ) ) ).
% s.prod_unit_l
thf(fact_85_s_OUnits__inv__comm,axiom,
! [X: nat,Y: nat] :
( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Y @ X )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) ) ) ).
% s.Units_inv_comm
thf(fact_86_s_Oproperfactor__unitE,axiom,
! [U: nat,A: nat] :
( ( member_nat @ U @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( proper2699154196232879280t_unit @ ( mod_ring @ n ) @ A @ U )
=> ~ ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.properfactor_unitE
thf(fact_87_s_OUnits__r__inv__ex,axiom,
! [X: nat] :
( ( member_nat @ X @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
& ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ X4 )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.Units_r_inv_ex
thf(fact_88_s_OUnits__l__inv__ex,axiom,
! [X: nat] :
( ( member_nat @ X @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
& ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X4 @ X )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.Units_l_inv_ex
thf(fact_89_s_Oadd_Oright__cancel,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ X )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% s.add.right_cancel
thf(fact_90_s_Oadd_Om__closed,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.add.m_closed
thf(fact_91_s_Oadd_Onat__pow__closed,axiom,
! [X: nat,N: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.add.nat_pow_closed
thf(fact_92_s_Oone__closed,axiom,
member_nat @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ).
% s.one_closed
thf(fact_93_s_Oadd_Oint__pow__closed,axiom,
! [X: nat,I: int] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ X ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.add.int_pow_closed
thf(fact_94_s_OUnits__m__closed,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y ) @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.Units_m_closed
thf(fact_95_s_OUnits__one__closed,axiom,
member_nat @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) ).
% s.Units_one_closed
thf(fact_96_s_Or__one,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
= X ) ) ).
% s.r_one
thf(fact_97_s_Ol__one,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) @ X )
= X ) ) ).
% s.l_one
thf(fact_98_s_OUnits__l__cancel,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member_nat @ X @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y )
= ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% s.Units_l_cancel
thf(fact_99_monoid__cancel_Ois__monoid__cancel,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( monoid2713784563552164055t_unit @ G2 )
=> ( monoid2713784563552164055t_unit @ G2 ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_100_monoid__cancel_Oassoc__unit__l,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( monoid497721730651901107t_unit @ G2 )
=> ( ( associ3816808824722140549t_unit @ G2 @ A @ B )
=> ( ( member_set_int @ B @ ( units_4038138251425117394t_unit @ G2 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( member_set_int @ A @ ( units_4038138251425117394t_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_101_monoid__cancel_Oassoc__unit__l,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( monoid2713784563552164055t_unit @ G2 )
=> ( ( associ4357985397585971625t_unit @ G2 @ A @ B )
=> ( ( member_nat @ B @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( member_nat @ A @ ( units_1295200668666280182t_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_102_monoid__cancel_Oassoc__unit__r,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( monoid497721730651901107t_unit @ G2 )
=> ( ( member_set_int @ A @ ( units_4038138251425117394t_unit @ G2 ) )
=> ( ( associ3816808824722140549t_unit @ G2 @ A @ B )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( member_set_int @ B @ ( units_4038138251425117394t_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_103_monoid__cancel_Oassoc__unit__r,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( monoid2713784563552164055t_unit @ G2 )
=> ( ( member_nat @ A @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( associ4357985397585971625t_unit @ G2 @ A @ B )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( member_nat @ B @ ( units_1295200668666280182t_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_104_ring__iso__memE_I4_J,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit] :
( ( member_nat_nat @ H @ ( ring_i3628375019190340560t_unit @ R2 @ S ) )
=> ( ( H @ ( one_na902338870878123981t_unit @ R2 ) )
= ( one_na902338870878123981t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_105_ring__iso__memE_I4_J,axiom,
! [H: nat > set_int,R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit] :
( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R2 @ S ) )
=> ( ( H @ ( one_na902338870878123981t_unit @ R2 ) )
= ( one_se8065767436706823081t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_106_monoid__cancel_OassociatedD2,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( monoid497721730651901107t_unit @ G2 )
=> ( ( associ3816808824722140549t_unit @ G2 @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ ( units_4038138251425117394t_unit @ G2 ) )
& ( A
= ( mult_s3864001451298473021t_unit @ G2 @ B @ X4 ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_107_monoid__cancel_OassociatedD2,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( monoid2713784563552164055t_unit @ G2 )
=> ( ( associ4357985397585971625t_unit @ G2 @ A @ B )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( units_1295200668666280182t_unit @ G2 ) )
& ( A
= ( mult_n6028127365542633569t_unit @ G2 @ B @ X4 ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_108_monoid__cancel_OassociatedE2,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( monoid497721730651901107t_unit @ G2 )
=> ( ( associ3816808824722140549t_unit @ G2 @ A @ B )
=> ( ! [U2: set_int] :
( ( A
= ( mult_s3864001451298473021t_unit @ G2 @ B @ U2 ) )
=> ~ ( member_set_int @ U2 @ ( units_4038138251425117394t_unit @ G2 ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ~ ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_109_monoid__cancel_OassociatedE2,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( monoid2713784563552164055t_unit @ G2 )
=> ( ( associ4357985397585971625t_unit @ G2 @ A @ B )
=> ( ! [U2: nat] :
( ( A
= ( mult_n6028127365542633569t_unit @ G2 @ B @ U2 ) )
=> ~ ( member_nat @ U2 @ ( units_1295200668666280182t_unit @ G2 ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ~ ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_110_monoid__cancel_Oassociated__iff,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( monoid497721730651901107t_unit @ G2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( associ3816808824722140549t_unit @ G2 @ A @ B )
= ( ? [X3: set_int] :
( ( member_set_int @ X3 @ ( units_4038138251425117394t_unit @ G2 ) )
& ( A
= ( mult_s3864001451298473021t_unit @ G2 @ B @ X3 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_111_monoid__cancel_Oassociated__iff,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( monoid2713784563552164055t_unit @ G2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( associ4357985397585971625t_unit @ G2 @ A @ B )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( units_1295200668666280182t_unit @ G2 ) )
& ( A
= ( mult_n6028127365542633569t_unit @ G2 @ B @ X3 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_112_monoid__cancel_Oaxioms_I1_J,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( monoid2713784563552164055t_unit @ G2 )
=> ( monoid4477821099152670945t_unit @ G2 ) ) ).
% monoid_cancel.axioms(1)
thf(fact_113_monoid_OUnits__assoc,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ A @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( associ4357985397585971625t_unit @ G2 @ A @ B ) ) ) ) ).
% monoid.Units_assoc
thf(fact_114_ring__iso__memE_I3_J,axiom,
! [H: set_int > nat,R2: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,X: set_int,Y: set_int] :
( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ R2 @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( H @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) )
= ( add_nat_Product_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_115_ring__iso__memE_I3_J,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( member_nat_nat @ H @ ( ring_i3628375019190340560t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( add_nat_Product_unit @ R2 @ X @ Y ) )
= ( add_nat_Product_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_116_ring__iso__memE_I3_J,axiom,
! [H: nat > set_int,R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,X: nat,Y: nat] :
( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( add_nat_Product_unit @ R2 @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_117_monoid__cancel_Or__cancel,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,C: set_int,B: set_int] :
( ( monoid497721730651901107t_unit @ G2 )
=> ( ( ( mult_s3864001451298473021t_unit @ G2 @ A @ C )
= ( mult_s3864001451298473021t_unit @ G2 @ B @ C ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G2 ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_118_monoid__cancel_Or__cancel,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,C: nat,B: nat] :
( ( monoid2713784563552164055t_unit @ G2 )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ A @ C )
= ( mult_n6028127365542633569t_unit @ G2 @ B @ C ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_119_monoid__cancel_Ol__cancel,axiom,
! [G2: partia4934656038542163276t_unit,C: set_int,A: set_int,B: set_int] :
( ( monoid497721730651901107t_unit @ G2 )
=> ( ( ( mult_s3864001451298473021t_unit @ G2 @ C @ A )
= ( mult_s3864001451298473021t_unit @ G2 @ C @ B ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G2 ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_120_monoid__cancel_Ol__cancel,axiom,
! [G2: partia4692342223508353374t_unit,C: nat,A: nat,B: nat] :
( ( monoid2713784563552164055t_unit @ G2 )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ C @ A )
= ( mult_n6028127365542633569t_unit @ G2 @ C @ B ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_121_monoid_Oprod__unit__l,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ ( mult_s3864001451298473021t_unit @ G2 @ A @ B ) @ ( units_4038138251425117394t_unit @ G2 ) )
=> ( ( member_set_int @ A @ ( units_4038138251425117394t_unit @ G2 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( member_set_int @ B @ ( units_4038138251425117394t_unit @ G2 ) ) ) ) ) ) ) ).
% monoid.prod_unit_l
thf(fact_122_monoid_Oprod__unit__l,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ ( mult_n6028127365542633569t_unit @ G2 @ A @ B ) @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( member_nat @ A @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( member_nat @ B @ ( units_1295200668666280182t_unit @ G2 ) ) ) ) ) ) ) ).
% monoid.prod_unit_l
thf(fact_123_monoid_Oprod__unit__r,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ ( mult_s3864001451298473021t_unit @ G2 @ A @ B ) @ ( units_4038138251425117394t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( units_4038138251425117394t_unit @ G2 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( member_set_int @ A @ ( units_4038138251425117394t_unit @ G2 ) ) ) ) ) ) ) ).
% monoid.prod_unit_r
thf(fact_124_monoid_Oprod__unit__r,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ ( mult_n6028127365542633569t_unit @ G2 @ A @ B ) @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( member_nat @ A @ ( units_1295200668666280182t_unit @ G2 ) ) ) ) ) ) ) ).
% monoid.prod_unit_r
thf(fact_125_monoid_Oproperfactor__unitE,axiom,
! [G2: partia4934656038542163276t_unit,U: set_int,A: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ U @ ( units_4038138251425117394t_unit @ G2 ) )
=> ( ( proper1002977052347345036t_unit @ G2 @ A @ U )
=> ~ ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) ) ) ) ) ).
% monoid.properfactor_unitE
thf(fact_126_monoid_Oproperfactor__unitE,axiom,
! [G2: partia4692342223508353374t_unit,U: nat,A: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ U @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( proper2699154196232879280t_unit @ G2 @ A @ U )
=> ~ ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) ) ) ) ) ).
% monoid.properfactor_unitE
thf(fact_127_monoid_OassociatedI2,axiom,
! [G2: partia4934656038542163276t_unit,U: set_int,A: set_int,B: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ U @ ( units_4038138251425117394t_unit @ G2 ) )
=> ( ( A
= ( mult_s3864001451298473021t_unit @ G2 @ B @ U ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( associ3816808824722140549t_unit @ G2 @ A @ B ) ) ) ) ) ).
% monoid.associatedI2
thf(fact_128_monoid_OassociatedI2,axiom,
! [G2: partia4692342223508353374t_unit,U: nat,A: nat,B: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ U @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( A
= ( mult_n6028127365542633569t_unit @ G2 @ B @ U ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( associ4357985397585971625t_unit @ G2 @ A @ B ) ) ) ) ) ).
% monoid.associatedI2
thf(fact_129_monoid_OassociatedI2_H,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int,U: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( A
= ( mult_s3864001451298473021t_unit @ G2 @ B @ U ) )
=> ( ( member_set_int @ U @ ( units_4038138251425117394t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( associ3816808824722140549t_unit @ G2 @ A @ B ) ) ) ) ) ).
% monoid.associatedI2'
thf(fact_130_monoid_OassociatedI2_H,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat,U: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( A
= ( mult_n6028127365542633569t_unit @ G2 @ B @ U ) )
=> ( ( member_nat @ U @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( associ4357985397585971625t_unit @ G2 @ A @ B ) ) ) ) ) ).
% monoid.associatedI2'
thf(fact_131_monoid__cancel_Oassoc__l__cancel,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int,B3: set_int] :
( ( monoid497721730651901107t_unit @ G2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B3 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( associ3816808824722140549t_unit @ G2 @ ( mult_s3864001451298473021t_unit @ G2 @ A @ B ) @ ( mult_s3864001451298473021t_unit @ G2 @ A @ B3 ) )
=> ( associ3816808824722140549t_unit @ G2 @ B @ B3 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_132_monoid__cancel_Oassoc__l__cancel,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat,B3: nat] :
( ( monoid2713784563552164055t_unit @ G2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B3 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( associ4357985397585971625t_unit @ G2 @ ( mult_n6028127365542633569t_unit @ G2 @ A @ B ) @ ( mult_n6028127365542633569t_unit @ G2 @ A @ B3 ) )
=> ( associ4357985397585971625t_unit @ G2 @ B @ B3 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_133_monoid_Omonoid__cancelI,axiom,
! [G2: partia4934656038542163276t_unit] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ! [A2: set_int,B2: set_int,C2: set_int] :
( ( ( mult_s3864001451298473021t_unit @ G2 @ C2 @ A2 )
= ( mult_s3864001451298473021t_unit @ G2 @ C2 @ B2 ) )
=> ( ( member_set_int @ A2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ C2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: set_int,B2: set_int,C2: set_int] :
( ( ( mult_s3864001451298473021t_unit @ G2 @ A2 @ C2 )
= ( mult_s3864001451298473021t_unit @ G2 @ B2 @ C2 ) )
=> ( ( member_set_int @ A2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ C2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid497721730651901107t_unit @ G2 ) ) ) ) ).
% monoid.monoid_cancelI
thf(fact_134_monoid_Omonoid__cancelI,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ! [A2: nat,B2: nat,C2: nat] :
( ( ( mult_n6028127365542633569t_unit @ G2 @ C2 @ A2 )
= ( mult_n6028127365542633569t_unit @ G2 @ C2 @ B2 ) )
=> ( ( member_nat @ A2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ C2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: nat,B2: nat,C2: nat] :
( ( ( mult_n6028127365542633569t_unit @ G2 @ A2 @ C2 )
= ( mult_n6028127365542633569t_unit @ G2 @ B2 @ C2 ) )
=> ( ( member_nat @ A2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ C2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid2713784563552164055t_unit @ G2 ) ) ) ) ).
% monoid.monoid_cancelI
thf(fact_135_monoid__cancel_Oproperfactor__mult__lI,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int,C: set_int] :
( ( monoid497721730651901107t_unit @ G2 )
=> ( ( proper1002977052347345036t_unit @ G2 @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G2 ) )
=> ( proper1002977052347345036t_unit @ G2 @ ( mult_s3864001451298473021t_unit @ G2 @ C @ A ) @ ( mult_s3864001451298473021t_unit @ G2 @ C @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_lI
thf(fact_136_monoid__cancel_Oproperfactor__mult__lI,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat,C: nat] :
( ( monoid2713784563552164055t_unit @ G2 )
=> ( ( proper2699154196232879280t_unit @ G2 @ A @ B )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( proper2699154196232879280t_unit @ G2 @ ( mult_n6028127365542633569t_unit @ G2 @ C @ A ) @ ( mult_n6028127365542633569t_unit @ G2 @ C @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_lI
thf(fact_137_monoid__cancel_Oproperfactor__mult__l,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int,C: set_int] :
( ( monoid497721730651901107t_unit @ G2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( proper1002977052347345036t_unit @ G2 @ ( mult_s3864001451298473021t_unit @ G2 @ C @ A ) @ ( mult_s3864001451298473021t_unit @ G2 @ C @ B ) )
= ( proper1002977052347345036t_unit @ G2 @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_l
thf(fact_138_monoid__cancel_Oproperfactor__mult__l,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat,C: nat] :
( ( monoid2713784563552164055t_unit @ G2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( proper2699154196232879280t_unit @ G2 @ ( mult_n6028127365542633569t_unit @ G2 @ C @ A ) @ ( mult_n6028127365542633569t_unit @ G2 @ C @ B ) )
= ( proper2699154196232879280t_unit @ G2 @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_l
thf(fact_139_monoid_Ol__one,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( mult_s3864001451298473021t_unit @ G2 @ ( one_se8065767436706823081t_unit @ G2 ) @ X )
= X ) ) ) ).
% monoid.l_one
thf(fact_140_monoid_Ol__one,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ ( one_na902338870878123981t_unit @ G2 ) @ X )
= X ) ) ) ).
% monoid.l_one
thf(fact_141_monoid_Or__one,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( mult_s3864001451298473021t_unit @ G2 @ X @ ( one_se8065767436706823081t_unit @ G2 ) )
= X ) ) ) ).
% monoid.r_one
thf(fact_142_monoid_Or__one,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ X @ ( one_na902338870878123981t_unit @ G2 ) )
= X ) ) ) ).
% monoid.r_one
thf(fact_143_monoid_Oone__closed,axiom,
! [G2: partia4934656038542163276t_unit] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ G2 ) @ ( partia966996272515721803t_unit @ G2 ) ) ) ).
% monoid.one_closed
thf(fact_144_monoid_Oone__closed,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ G2 ) @ ( partia3499330772048238685t_unit @ G2 ) ) ) ).
% monoid.one_closed
thf(fact_145_ringI,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( abelia23968383328945916t_unit @ R2 )
=> ( ( monoid833175047693460669t_unit @ R2 )
=> ( ! [X4: set_int,Y3: set_int,Z2: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( add_se5859248395121729892t_unit @ R2 @ X4 @ Y3 ) @ Z2 )
= ( add_se5859248395121729892t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ X4 @ Z2 ) @ ( mult_s3864001451298473021t_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: set_int,Y3: set_int,Z2: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ Z2 @ ( add_se5859248395121729892t_unit @ R2 @ X4 @ Y3 ) )
= ( add_se5859248395121729892t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ Z2 @ X4 ) @ ( mult_s3864001451298473021t_unit @ R2 @ Z2 @ Y3 ) ) ) ) ) )
=> ( ring_s5316885176909347197t_unit @ R2 ) ) ) ) ) ).
% ringI
thf(fact_146_ringI,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( abelia406319425121669262t_unit @ R2 )
=> ( ( monoid4477821099152670945t_unit @ R2 )
=> ( ! [X4: nat,Y3: nat,Z2: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( add_nat_Product_unit @ R2 @ X4 @ Y3 ) @ Z2 )
= ( add_nat_Product_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ X4 @ Z2 ) @ ( mult_n6028127365542633569t_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: nat,Y3: nat,Z2: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ Z2 @ ( add_nat_Product_unit @ R2 @ X4 @ Y3 ) )
= ( add_nat_Product_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ Z2 @ X4 ) @ ( mult_n6028127365542633569t_unit @ R2 @ Z2 @ Y3 ) ) ) ) ) )
=> ( ring_n9194430563101542159t_unit @ R2 ) ) ) ) ) ).
% ringI
thf(fact_147_ring__hom__memI,axiom,
! [R2: partia4934656038542163276t_unit,H: set_int > set_int,S: partia4934656038542163276t_unit] :
( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( H @ X4 ) @ ( partia966996272515721803t_unit @ S ) ) )
=> ( ! [X4: set_int,Y3: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( H @ ( mult_s3864001451298473021t_unit @ R2 @ X4 @ Y3 ) )
= ( mult_s3864001451298473021t_unit @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ! [X4: set_int,Y3: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( H @ ( add_se5859248395121729892t_unit @ R2 @ X4 @ Y3 ) )
= ( add_se5859248395121729892t_unit @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ( ( H @ ( one_se8065767436706823081t_unit @ R2 ) )
= ( one_se8065767436706823081t_unit @ S ) )
=> ( member5205197933313416826et_int @ H @ ( ring_h3404898052528352314t_unit @ R2 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_148_ring__hom__memI,axiom,
! [R2: partia4934656038542163276t_unit,H: set_int > nat,S: partia4692342223508353374t_unit] :
( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_nat @ ( H @ X4 ) @ ( partia3499330772048238685t_unit @ S ) ) )
=> ( ! [X4: set_int,Y3: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( H @ ( mult_s3864001451298473021t_unit @ R2 @ X4 @ Y3 ) )
= ( mult_n6028127365542633569t_unit @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ! [X4: set_int,Y3: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( H @ ( add_se5859248395121729892t_unit @ R2 @ X4 @ Y3 ) )
= ( add_nat_Product_unit @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ( ( H @ ( one_se8065767436706823081t_unit @ R2 ) )
= ( one_na902338870878123981t_unit @ S ) )
=> ( member_set_int_nat @ H @ ( ring_h9051218956466089420t_unit @ R2 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_149_ring__hom__memI,axiom,
! [R2: partia4692342223508353374t_unit,H: nat > set_int,S: partia4934656038542163276t_unit] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_set_int @ ( H @ X4 ) @ ( partia966996272515721803t_unit @ S ) ) )
=> ( ! [X4: nat,Y3: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( mult_n6028127365542633569t_unit @ R2 @ X4 @ Y3 ) )
= ( mult_s3864001451298473021t_unit @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ! [X4: nat,Y3: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( add_nat_Product_unit @ R2 @ X4 @ Y3 ) )
= ( add_se5859248395121729892t_unit @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ( ( H @ ( one_na902338870878123981t_unit @ R2 ) )
= ( one_se8065767436706823081t_unit @ S ) )
=> ( member_nat_set_int @ H @ ( ring_h4752909569380436264t_unit @ R2 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_150_ring__hom__memI,axiom,
! [R2: partia4692342223508353374t_unit,H: nat > nat,S: partia4692342223508353374t_unit] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( H @ X4 ) @ ( partia3499330772048238685t_unit @ S ) ) )
=> ( ! [X4: nat,Y3: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( mult_n6028127365542633569t_unit @ R2 @ X4 @ Y3 ) )
= ( mult_n6028127365542633569t_unit @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ! [X4: nat,Y3: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( add_nat_Product_unit @ R2 @ X4 @ Y3 ) )
= ( add_nat_Product_unit @ S @ ( H @ X4 ) @ ( H @ Y3 ) ) ) ) )
=> ( ( ( H @ ( one_na902338870878123981t_unit @ R2 ) )
= ( one_na902338870878123981t_unit @ S ) )
=> ( member_nat_nat @ H @ ( ring_h4412161176302437050t_unit @ R2 @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_151_monoid_OUnits__r__inv__ex,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ X @ ( units_4038138251425117394t_unit @ G2 ) )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
& ( ( mult_s3864001451298473021t_unit @ G2 @ X @ X4 )
= ( one_se8065767436706823081t_unit @ G2 ) ) ) ) ) ).
% monoid.Units_r_inv_ex
thf(fact_152_monoid_OUnits__r__inv__ex,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1295200668666280182t_unit @ G2 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
& ( ( mult_n6028127365542633569t_unit @ G2 @ X @ X4 )
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ) ).
% monoid.Units_r_inv_ex
thf(fact_153_monoid_OUnits__l__inv__ex,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ X @ ( units_4038138251425117394t_unit @ G2 ) )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
& ( ( mult_s3864001451298473021t_unit @ G2 @ X4 @ X )
= ( one_se8065767436706823081t_unit @ G2 ) ) ) ) ) ).
% monoid.Units_l_inv_ex
thf(fact_154_monoid_OUnits__l__inv__ex,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1295200668666280182t_unit @ G2 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
& ( ( mult_n6028127365542633569t_unit @ G2 @ X4 @ X )
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ) ).
% monoid.Units_l_inv_ex
thf(fact_155_s_Oline__extension__mem__iff,axiom,
! [U: nat,K2: set_nat,A: nat,E: set_nat] :
( ( member_nat @ U @ ( embedd838748496991043025t_unit @ ( mod_ring @ n ) @ K2 @ A @ E ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ K2 )
& ? [Y4: nat] :
( ( member_nat @ Y4 @ E )
& ( U
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X3 @ A ) @ Y4 ) ) ) ) ) ) ).
% s.line_extension_mem_iff
thf(fact_156_s_Oirreducible__prod__rI,axiom,
! [A: nat,B: nat] :
( ( irredu2811410805266234189t_unit @ ( mod_ring @ n ) @ A )
=> ( ( member_nat @ B @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( irredu2811410805266234189t_unit @ ( mod_ring @ n ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ A @ B ) ) ) ) ) ) ).
% s.irreducible_prod_rI
thf(fact_157_s_Oadd_Oint__pow__mult,axiom,
! [X: nat,I: int,J: int] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ ( plus_plus_int @ I @ J ) @ X )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ X ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ J @ X ) ) ) ) ).
% s.add.int_pow_mult
thf(fact_158_s_Oadd_Onat__pow__mult,axiom,
! [X: nat,N: nat,M: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ M @ X ) )
= ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ ( plus_plus_nat @ N @ M ) @ X ) ) ) ).
% s.add.nat_pow_mult
thf(fact_159_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_160_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_161_Divisibility_OirreducibleD,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( irredu6406935661358989097t_unit @ G2 @ A )
=> ( ( proper1002977052347345036t_unit @ G2 @ B @ A )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( member_set_int @ B @ ( units_4038138251425117394t_unit @ G2 ) ) ) ) ) ).
% Divisibility.irreducibleD
thf(fact_162_Divisibility_OirreducibleD,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( irredu2811410805266234189t_unit @ G2 @ A )
=> ( ( proper2699154196232879280t_unit @ G2 @ B @ A )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( member_nat @ B @ ( units_1295200668666280182t_unit @ G2 ) ) ) ) ) ).
% Divisibility.irreducibleD
thf(fact_163_irreducibleE,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int] :
( ( irredu6406935661358989097t_unit @ G2 @ A )
=> ~ ( ~ ( member_set_int @ A @ ( units_4038138251425117394t_unit @ G2 ) )
=> ~ ! [B4: set_int] :
( ( ( member_set_int @ B4 @ ( partia966996272515721803t_unit @ G2 ) )
& ( proper1002977052347345036t_unit @ G2 @ B4 @ A ) )
=> ( member_set_int @ B4 @ ( units_4038138251425117394t_unit @ G2 ) ) ) ) ) ).
% irreducibleE
thf(fact_164_irreducibleE,axiom,
! [G2: partia4692342223508353374t_unit,A: nat] :
( ( irredu2811410805266234189t_unit @ G2 @ A )
=> ~ ( ~ ( member_nat @ A @ ( units_1295200668666280182t_unit @ G2 ) )
=> ~ ! [B4: nat] :
( ( ( member_nat @ B4 @ ( partia3499330772048238685t_unit @ G2 ) )
& ( proper2699154196232879280t_unit @ G2 @ B4 @ A ) )
=> ( member_nat @ B4 @ ( units_1295200668666280182t_unit @ G2 ) ) ) ) ) ).
% irreducibleE
thf(fact_165_Divisibility_OirreducibleI,axiom,
! [A: set_int,G2: partia4934656038542163276t_unit] :
( ~ ( member_set_int @ A @ ( units_4038138251425117394t_unit @ G2 ) )
=> ( ! [B2: set_int] :
( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( proper1002977052347345036t_unit @ G2 @ B2 @ A )
=> ( member_set_int @ B2 @ ( units_4038138251425117394t_unit @ G2 ) ) ) )
=> ( irredu6406935661358989097t_unit @ G2 @ A ) ) ) ).
% Divisibility.irreducibleI
thf(fact_166_Divisibility_OirreducibleI,axiom,
! [A: nat,G2: partia4692342223508353374t_unit] :
( ~ ( member_nat @ A @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( proper2699154196232879280t_unit @ G2 @ B2 @ A )
=> ( member_nat @ B2 @ ( units_1295200668666280182t_unit @ G2 ) ) ) )
=> ( irredu2811410805266234189t_unit @ G2 @ A ) ) ) ).
% Divisibility.irreducibleI
thf(fact_167_Divisibility_Oirreducible__def,axiom,
( irredu6406935661358989097t_unit
= ( ^ [G3: partia4934656038542163276t_unit,A4: set_int] :
( ~ ( member_set_int @ A4 @ ( units_4038138251425117394t_unit @ G3 ) )
& ! [X3: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ G3 ) )
=> ( ( proper1002977052347345036t_unit @ G3 @ X3 @ A4 )
=> ( member_set_int @ X3 @ ( units_4038138251425117394t_unit @ G3 ) ) ) ) ) ) ) ).
% Divisibility.irreducible_def
thf(fact_168_Divisibility_Oirreducible__def,axiom,
( irredu2811410805266234189t_unit
= ( ^ [G3: partia4692342223508353374t_unit,A4: nat] :
( ~ ( member_nat @ A4 @ ( units_1295200668666280182t_unit @ G3 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( partia3499330772048238685t_unit @ G3 ) )
=> ( ( proper2699154196232879280t_unit @ G3 @ X3 @ A4 )
=> ( member_nat @ X3 @ ( units_1295200668666280182t_unit @ G3 ) ) ) ) ) ) ) ).
% Divisibility.irreducible_def
thf(fact_169_monoid__cancel_Oirreducible__cong,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,A5: set_int] :
( ( monoid497721730651901107t_unit @ G2 )
=> ( ( irredu6406935661358989097t_unit @ G2 @ A )
=> ( ( associ3816808824722140549t_unit @ G2 @ A @ A5 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ A5 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( irredu6406935661358989097t_unit @ G2 @ A5 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_170_monoid__cancel_Oirreducible__cong,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,A5: nat] :
( ( monoid2713784563552164055t_unit @ G2 )
=> ( ( irredu2811410805266234189t_unit @ G2 @ A )
=> ( ( associ4357985397585971625t_unit @ G2 @ A @ A5 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ A5 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( irredu2811410805266234189t_unit @ G2 @ A5 ) ) ) ) ) ) ).
% monoid_cancel.irreducible_cong
thf(fact_171_monoid_Oirreducible__prod__rI,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( irredu6406935661358989097t_unit @ G2 @ A )
=> ( ( member_set_int @ B @ ( units_4038138251425117394t_unit @ G2 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( irredu6406935661358989097t_unit @ G2 @ ( mult_s3864001451298473021t_unit @ G2 @ A @ B ) ) ) ) ) ) ) ).
% monoid.irreducible_prod_rI
thf(fact_172_monoid_Oirreducible__prod__rI,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( irredu2811410805266234189t_unit @ G2 @ A )
=> ( ( member_nat @ B @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( irredu2811410805266234189t_unit @ G2 @ ( mult_n6028127365542633569t_unit @ G2 @ A @ B ) ) ) ) ) ) ) ).
% monoid.irreducible_prod_rI
thf(fact_173_ring__hom__closed,axiom,
! [H: set_int > set_int,R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,X: set_int] :
( ( member5205197933313416826et_int @ H @ ( ring_h3404898052528352314t_unit @ R2 @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( H @ X ) @ ( partia966996272515721803t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_174_ring__hom__closed,axiom,
! [H: set_int > nat,R2: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,X: set_int] :
( ( member_set_int_nat @ H @ ( ring_h9051218956466089420t_unit @ R2 @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_nat @ ( H @ X ) @ ( partia3499330772048238685t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_175_ring__hom__closed,axiom,
! [H: nat > set_int,R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,X: nat] :
( ( member_nat_set_int @ H @ ( ring_h4752909569380436264t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_set_int @ ( H @ X ) @ ( partia966996272515721803t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_176_ring__hom__closed,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit,X: nat] :
( ( member_nat_nat @ H @ ( ring_h4412161176302437050t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( H @ X ) @ ( partia3499330772048238685t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_177_ring_Ois__abelian__group,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( abelia23968383328945916t_unit @ R2 ) ) ).
% ring.is_abelian_group
thf(fact_178_ring_Ois__abelian__group,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( abelia406319425121669262t_unit @ R2 ) ) ).
% ring.is_abelian_group
thf(fact_179_ring__hom__one,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit] :
( ( member_nat_nat @ H @ ( ring_h4412161176302437050t_unit @ R2 @ S ) )
=> ( ( H @ ( one_na902338870878123981t_unit @ R2 ) )
= ( one_na902338870878123981t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_180_abelian__group_Oaxioms_I1_J,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( abelia362511065248671243t_unit @ G2 ) ) ).
% abelian_group.axioms(1)
thf(fact_181_ring_Ois__monoid,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( monoid833175047693460669t_unit @ R2 ) ) ).
% ring.is_monoid
thf(fact_182_ring_Ois__monoid,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( monoid4477821099152670945t_unit @ R2 ) ) ).
% ring.is_monoid
thf(fact_183_semiring_Oaxioms_I1_J,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( abelia362511065248671243t_unit @ R2 ) ) ).
% semiring.axioms(1)
thf(fact_184_semiring_Oaxioms_I2_J,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( monoid4477821099152670945t_unit @ R2 ) ) ).
% semiring.axioms(2)
thf(fact_185_ring_Oring__simprules_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_186_ring_Oring__simprules_I1_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ X @ Y ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_187_ring_Oring__simprules_I7_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R2 @ X @ ( add_se5859248395121729892t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_188_ring_Oring__simprules_I7_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( add_nat_Product_unit @ R2 @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ R2 @ X @ ( add_nat_Product_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_189_ring_Oring__simprules_I10_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ Y )
= ( add_se5859248395121729892t_unit @ R2 @ Y @ X ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_190_ring_Oring__simprules_I10_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ Y )
= ( add_nat_Product_unit @ R2 @ Y @ X ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_191_ring_Oring__simprules_I22_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ ( add_se5859248395121729892t_unit @ R2 @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ R2 @ Y @ ( add_se5859248395121729892t_unit @ R2 @ X @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_192_ring_Oring__simprules_I22_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ ( add_nat_Product_unit @ R2 @ Y @ Z ) )
= ( add_nat_Product_unit @ R2 @ Y @ ( add_nat_Product_unit @ R2 @ X @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_193_ring_Oring__simprules_I5_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Y ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_194_ring_Oring__simprules_I5_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_195_ring_Oring__simprules_I11_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ R2 @ X @ ( mult_s3864001451298473021t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_196_ring_Oring__simprules_I11_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) @ Z )
= ( mult_n6028127365542633569t_unit @ R2 @ X @ ( mult_n6028127365542633569t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_197_monoid_Ocarrier__not__empty,axiom,
! [G2: partia4934656038542163276t_unit] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( partia966996272515721803t_unit @ G2 )
!= bot_bot_set_set_int ) ) ).
% monoid.carrier_not_empty
thf(fact_198_monoid_Ocarrier__not__empty,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( partia3499330772048238685t_unit @ G2 )
!= bot_bot_set_nat ) ) ).
% monoid.carrier_not_empty
thf(fact_199_ring_Oring__simprules_I6_J,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ R2 ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% ring.ring_simprules(6)
thf(fact_200_ring_Oring__simprules_I6_J,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ R2 ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% ring.ring_simprules(6)
thf(fact_201_monoid_Om__closed,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ G2 @ X @ Y ) @ ( partia966996272515721803t_unit @ G2 ) ) ) ) ) ).
% monoid.m_closed
thf(fact_202_monoid_Om__closed,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ G2 @ X @ Y ) @ ( partia3499330772048238685t_unit @ G2 ) ) ) ) ) ).
% monoid.m_closed
thf(fact_203_monoid_Om__assoc,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( mult_s3864001451298473021t_unit @ G2 @ ( mult_s3864001451298473021t_unit @ G2 @ X @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ G2 @ X @ ( mult_s3864001451298473021t_unit @ G2 @ Y @ Z ) ) ) ) ) ) ) ).
% monoid.m_assoc
thf(fact_204_monoid_Om__assoc,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ ( mult_n6028127365542633569t_unit @ G2 @ X @ Y ) @ Z )
= ( mult_n6028127365542633569t_unit @ G2 @ X @ ( mult_n6028127365542633569t_unit @ G2 @ Y @ Z ) ) ) ) ) ) ) ).
% monoid.m_assoc
thf(fact_205_monoid_OUnits__closed,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ X @ ( units_4038138251425117394t_unit @ G2 ) )
=> ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) ) ) ) ).
% monoid.Units_closed
thf(fact_206_monoid_OUnits__closed,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) ) ) ) ).
% monoid.Units_closed
thf(fact_207_abelian__groupE_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_208_abelian__groupE_I1_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ X @ Y ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_209_abelian__groupE_I3_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( abelia23968383328945916t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R2 @ X @ ( add_se5859248395121729892t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_210_abelian__groupE_I3_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( abelia406319425121669262t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( add_nat_Product_unit @ R2 @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ R2 @ X @ ( add_nat_Product_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_211_abelian__groupE_I4_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ Y )
= ( add_se5859248395121729892t_unit @ R2 @ Y @ X ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_212_abelian__groupE_I4_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ Y )
= ( add_nat_Product_unit @ R2 @ Y @ X ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_213_ring__hom__add,axiom,
! [H: set_int > nat,R2: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,X: set_int,Y: set_int] :
( ( member_set_int_nat @ H @ ( ring_h9051218956466089420t_unit @ R2 @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( H @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) )
= ( add_nat_Product_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_214_ring__hom__add,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( member_nat_nat @ H @ ( ring_h4412161176302437050t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( add_nat_Product_unit @ R2 @ X @ Y ) )
= ( add_nat_Product_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_215_monoid_OUnits__m__closed,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ G2 @ X @ Y ) @ ( units_1295200668666280182t_unit @ G2 ) ) ) ) ) ).
% monoid.Units_m_closed
thf(fact_216_ring__hom__mult,axiom,
! [H: set_int > nat,R2: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,X: set_int,Y: set_int] :
( ( member_set_int_nat @ H @ ( ring_h9051218956466089420t_unit @ R2 @ S ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( H @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Y ) )
= ( mult_n6028127365542633569t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_217_ring__hom__mult,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( member_nat_nat @ H @ ( ring_h4412161176302437050t_unit @ R2 @ S ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( H @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) )
= ( mult_n6028127365542633569t_unit @ S @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_218_abelian__monoid_Oa__closed,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( abelia3815030880812984441t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ G2 @ X @ Y ) @ ( partia966996272515721803t_unit @ G2 ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_219_abelian__monoid_Oa__closed,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( abelia362511065248671243t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( member_nat @ ( add_nat_Product_unit @ G2 @ X @ Y ) @ ( partia3499330772048238685t_unit @ G2 ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_220_abelian__monoid_Oa__lcomm,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( abelia3815030880812984441t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( add_se5859248395121729892t_unit @ G2 @ X @ ( add_se5859248395121729892t_unit @ G2 @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ G2 @ Y @ ( add_se5859248395121729892t_unit @ G2 @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_221_abelian__monoid_Oa__lcomm,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( abelia362511065248671243t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( add_nat_Product_unit @ G2 @ X @ ( add_nat_Product_unit @ G2 @ Y @ Z ) )
= ( add_nat_Product_unit @ G2 @ Y @ ( add_nat_Product_unit @ G2 @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_222_abelian__monoid_Oa__assoc,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( abelia3815030880812984441t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( add_se5859248395121729892t_unit @ G2 @ ( add_se5859248395121729892t_unit @ G2 @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ G2 @ X @ ( add_se5859248395121729892t_unit @ G2 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_223_abelian__monoid_Oa__assoc,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( abelia362511065248671243t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( add_nat_Product_unit @ G2 @ ( add_nat_Product_unit @ G2 @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ G2 @ X @ ( add_nat_Product_unit @ G2 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_224_abelian__monoid_Oa__comm,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( abelia3815030880812984441t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( add_se5859248395121729892t_unit @ G2 @ X @ Y )
= ( add_se5859248395121729892t_unit @ G2 @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_225_abelian__monoid_Oa__comm,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( abelia362511065248671243t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( add_nat_Product_unit @ G2 @ X @ Y )
= ( add_nat_Product_unit @ G2 @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_226_abelian__monoidE_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( abelia3815030880812984441t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_227_abelian__monoidE_I1_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( abelia362511065248671243t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ X @ Y ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_228_abelian__monoidE_I3_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( abelia3815030880812984441t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R2 @ X @ ( add_se5859248395121729892t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_229_abelian__monoidE_I3_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( abelia362511065248671243t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( add_nat_Product_unit @ R2 @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ R2 @ X @ ( add_nat_Product_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_230_abelian__monoidE_I5_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( abelia3815030880812984441t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ Y )
= ( add_se5859248395121729892t_unit @ R2 @ Y @ X ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_231_abelian__monoidE_I5_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( abelia362511065248671243t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ Y )
= ( add_nat_Product_unit @ R2 @ Y @ X ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_232_monoid_OUnits__one__closed,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ G2 ) @ ( units_1295200668666280182t_unit @ G2 ) ) ) ).
% monoid.Units_one_closed
thf(fact_233_semiring_Osemiring__simprules_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_234_semiring_Osemiring__simprules_I1_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ X @ Y ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_235_semiring_Osemiring__simprules_I5_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R2 @ X @ ( add_se5859248395121729892t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_236_semiring_Osemiring__simprules_I5_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( add_nat_Product_unit @ R2 @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ R2 @ X @ ( add_nat_Product_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_237_semiring_Osemiring__simprules_I7_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ Y )
= ( add_se5859248395121729892t_unit @ R2 @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_238_semiring_Osemiring__simprules_I7_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ Y )
= ( add_nat_Product_unit @ R2 @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_239_semiring_Osemiring__simprules_I12_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ ( add_se5859248395121729892t_unit @ R2 @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ R2 @ Y @ ( add_se5859248395121729892t_unit @ R2 @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_240_semiring_Osemiring__simprules_I12_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ ( add_nat_Product_unit @ R2 @ Y @ Z ) )
= ( add_nat_Product_unit @ R2 @ Y @ ( add_nat_Product_unit @ R2 @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_241_semiring_Osemiring__simprules_I3_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Y ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_242_semiring_Osemiring__simprules_I3_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_243_semiring_Osemiring__simprules_I8_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ R2 @ X @ ( mult_s3864001451298473021t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_244_semiring_Osemiring__simprules_I8_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) @ Z )
= ( mult_n6028127365542633569t_unit @ R2 @ X @ ( mult_n6028127365542633569t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_245_semiring_Osemiring__simprules_I4_J,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ R2 ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_246_semiring_Osemiring__simprules_I4_J,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ R2 ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_247_ring_Oring__simprules_I13_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Z ) @ ( mult_s3864001451298473021t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_248_ring_Oring__simprules_I13_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( add_nat_Product_unit @ R2 @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Z ) @ ( mult_n6028127365542633569t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_249_ring_Oring__simprules_I23_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ Z @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ Z @ X ) @ ( mult_s3864001451298473021t_unit @ R2 @ Z @ Y ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_250_ring_Oring__simprules_I23_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ Z @ ( add_nat_Product_unit @ R2 @ X @ Y ) )
= ( add_nat_Product_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ Z @ X ) @ ( mult_n6028127365542633569t_unit @ R2 @ Z @ Y ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_251_ring_Oring__simprules_I12_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( one_se8065767436706823081t_unit @ R2 ) @ X )
= X ) ) ) ).
% ring.ring_simprules(12)
thf(fact_252_ring_Oring__simprules_I12_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( one_na902338870878123981t_unit @ R2 ) @ X )
= X ) ) ) ).
% ring.ring_simprules(12)
thf(fact_253_Group_Omonoid__def,axiom,
( monoid833175047693460669t_unit
= ( ^ [G3: partia4934656038542163276t_unit] :
( ! [X3: set_int,Y4: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ G3 ) )
=> ( ( member_set_int @ Y4 @ ( partia966996272515721803t_unit @ G3 ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ G3 @ X3 @ Y4 ) @ ( partia966996272515721803t_unit @ G3 ) ) ) )
& ! [X3: set_int,Y4: set_int,Z3: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ G3 ) )
=> ( ( member_set_int @ Y4 @ ( partia966996272515721803t_unit @ G3 ) )
=> ( ( member_set_int @ Z3 @ ( partia966996272515721803t_unit @ G3 ) )
=> ( ( mult_s3864001451298473021t_unit @ G3 @ ( mult_s3864001451298473021t_unit @ G3 @ X3 @ Y4 ) @ Z3 )
= ( mult_s3864001451298473021t_unit @ G3 @ X3 @ ( mult_s3864001451298473021t_unit @ G3 @ Y4 @ Z3 ) ) ) ) ) )
& ( member_set_int @ ( one_se8065767436706823081t_unit @ G3 ) @ ( partia966996272515721803t_unit @ G3 ) )
& ! [X3: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ G3 ) )
=> ( ( mult_s3864001451298473021t_unit @ G3 @ ( one_se8065767436706823081t_unit @ G3 ) @ X3 )
= X3 ) )
& ! [X3: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ G3 ) )
=> ( ( mult_s3864001451298473021t_unit @ G3 @ X3 @ ( one_se8065767436706823081t_unit @ G3 ) )
= X3 ) ) ) ) ) ).
% Group.monoid_def
thf(fact_254_Group_Omonoid__def,axiom,
( monoid4477821099152670945t_unit
= ( ^ [G3: partia4692342223508353374t_unit] :
( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ ( partia3499330772048238685t_unit @ G3 ) )
=> ( ( member_nat @ Y4 @ ( partia3499330772048238685t_unit @ G3 ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ G3 @ X3 @ Y4 ) @ ( partia3499330772048238685t_unit @ G3 ) ) ) )
& ! [X3: nat,Y4: nat,Z3: nat] :
( ( member_nat @ X3 @ ( partia3499330772048238685t_unit @ G3 ) )
=> ( ( member_nat @ Y4 @ ( partia3499330772048238685t_unit @ G3 ) )
=> ( ( member_nat @ Z3 @ ( partia3499330772048238685t_unit @ G3 ) )
=> ( ( mult_n6028127365542633569t_unit @ G3 @ ( mult_n6028127365542633569t_unit @ G3 @ X3 @ Y4 ) @ Z3 )
= ( mult_n6028127365542633569t_unit @ G3 @ X3 @ ( mult_n6028127365542633569t_unit @ G3 @ Y4 @ Z3 ) ) ) ) ) )
& ( member_nat @ ( one_na902338870878123981t_unit @ G3 ) @ ( partia3499330772048238685t_unit @ G3 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( partia3499330772048238685t_unit @ G3 ) )
=> ( ( mult_n6028127365542633569t_unit @ G3 @ ( one_na902338870878123981t_unit @ G3 ) @ X3 )
= X3 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( partia3499330772048238685t_unit @ G3 ) )
=> ( ( mult_n6028127365542633569t_unit @ G3 @ X3 @ ( one_na902338870878123981t_unit @ G3 ) )
= X3 ) ) ) ) ) ).
% Group.monoid_def
thf(fact_255_monoidI,axiom,
! [G2: partia4934656038542163276t_unit] :
( ! [X4: set_int,Y3: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ G2 @ X4 @ Y3 ) @ ( partia966996272515721803t_unit @ G2 ) ) ) )
=> ( ( member_set_int @ ( one_se8065767436706823081t_unit @ G2 ) @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ! [X4: set_int,Y3: set_int,Z2: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Z2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( mult_s3864001451298473021t_unit @ G2 @ ( mult_s3864001451298473021t_unit @ G2 @ X4 @ Y3 ) @ Z2 )
= ( mult_s3864001451298473021t_unit @ G2 @ X4 @ ( mult_s3864001451298473021t_unit @ G2 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( mult_s3864001451298473021t_unit @ G2 @ ( one_se8065767436706823081t_unit @ G2 ) @ X4 )
= X4 ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( mult_s3864001451298473021t_unit @ G2 @ X4 @ ( one_se8065767436706823081t_unit @ G2 ) )
= X4 ) )
=> ( monoid833175047693460669t_unit @ G2 ) ) ) ) ) ) ).
% monoidI
thf(fact_256_monoidI,axiom,
! [G2: partia4692342223508353374t_unit] :
( ! [X4: nat,Y3: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ G2 @ X4 @ Y3 ) @ ( partia3499330772048238685t_unit @ G2 ) ) ) )
=> ( ( member_nat @ ( one_na902338870878123981t_unit @ G2 ) @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ! [X4: nat,Y3: nat,Z2: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Z2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ ( mult_n6028127365542633569t_unit @ G2 @ X4 @ Y3 ) @ Z2 )
= ( mult_n6028127365542633569t_unit @ G2 @ X4 @ ( mult_n6028127365542633569t_unit @ G2 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ ( one_na902338870878123981t_unit @ G2 ) @ X4 )
= X4 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ X4 @ ( one_na902338870878123981t_unit @ G2 ) )
= X4 ) )
=> ( monoid4477821099152670945t_unit @ G2 ) ) ) ) ) ) ).
% monoidI
thf(fact_257_monoid_Oone__unique,axiom,
! [G2: partia4934656038542163276t_unit,U: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ U @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( mult_s3864001451298473021t_unit @ G2 @ U @ X4 )
= X4 ) )
=> ( U
= ( one_se8065767436706823081t_unit @ G2 ) ) ) ) ) ).
% monoid.one_unique
thf(fact_258_monoid_Oone__unique,axiom,
! [G2: partia4692342223508353374t_unit,U: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ U @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ U @ X4 )
= X4 ) )
=> ( U
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ) ).
% monoid.one_unique
thf(fact_259_monoid_Oinv__unique,axiom,
! [G2: partia4934656038542163276t_unit,Y: set_int,X: set_int,Y2: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( ( mult_s3864001451298473021t_unit @ G2 @ Y @ X )
= ( one_se8065767436706823081t_unit @ G2 ) )
=> ( ( ( mult_s3864001451298473021t_unit @ G2 @ X @ Y2 )
= ( one_se8065767436706823081t_unit @ G2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% monoid.inv_unique
thf(fact_260_monoid_Oinv__unique,axiom,
! [G2: partia4692342223508353374t_unit,Y: nat,X: nat,Y2: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ Y @ X )
= ( one_na902338870878123981t_unit @ G2 ) )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ X @ Y2 )
= ( one_na902338870878123981t_unit @ G2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% monoid.inv_unique
thf(fact_261_Group_Omonoid_Ointro,axiom,
! [G2: partia4934656038542163276t_unit] :
( ! [X4: set_int,Y3: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ G2 @ X4 @ Y3 ) @ ( partia966996272515721803t_unit @ G2 ) ) ) )
=> ( ! [X4: set_int,Y3: set_int,Z2: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Z2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( mult_s3864001451298473021t_unit @ G2 @ ( mult_s3864001451298473021t_unit @ G2 @ X4 @ Y3 ) @ Z2 )
= ( mult_s3864001451298473021t_unit @ G2 @ X4 @ ( mult_s3864001451298473021t_unit @ G2 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ( member_set_int @ ( one_se8065767436706823081t_unit @ G2 ) @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( mult_s3864001451298473021t_unit @ G2 @ ( one_se8065767436706823081t_unit @ G2 ) @ X4 )
= X4 ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( mult_s3864001451298473021t_unit @ G2 @ X4 @ ( one_se8065767436706823081t_unit @ G2 ) )
= X4 ) )
=> ( monoid833175047693460669t_unit @ G2 ) ) ) ) ) ) ).
% Group.monoid.intro
thf(fact_262_Group_Omonoid_Ointro,axiom,
! [G2: partia4692342223508353374t_unit] :
( ! [X4: nat,Y3: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ G2 @ X4 @ Y3 ) @ ( partia3499330772048238685t_unit @ G2 ) ) ) )
=> ( ! [X4: nat,Y3: nat,Z2: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Z2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ ( mult_n6028127365542633569t_unit @ G2 @ X4 @ Y3 ) @ Z2 )
= ( mult_n6028127365542633569t_unit @ G2 @ X4 @ ( mult_n6028127365542633569t_unit @ G2 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ( member_nat @ ( one_na902338870878123981t_unit @ G2 ) @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ ( one_na902338870878123981t_unit @ G2 ) @ X4 )
= X4 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ X4 @ ( one_na902338870878123981t_unit @ G2 ) )
= X4 ) )
=> ( monoid4477821099152670945t_unit @ G2 ) ) ) ) ) ) ).
% Group.monoid.intro
thf(fact_263_monoid_OUnits__l__cancel,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ X @ ( units_4038138251425117394t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( ( mult_s3864001451298473021t_unit @ G2 @ X @ Y )
= ( mult_s3864001451298473021t_unit @ G2 @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ).
% monoid.Units_l_cancel
thf(fact_264_monoid_OUnits__l__cancel,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ X @ Y )
= ( mult_n6028127365542633569t_unit @ G2 @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ) ).
% monoid.Units_l_cancel
thf(fact_265_ring_Oadd__pow__rdistr__int,axiom,
! [R2: partia4934656038542163276t_unit,A: set_int,B: set_int,K: int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ A @ ( add_po7581009264371422883it_int @ R2 @ K @ B ) )
= ( add_po7581009264371422883it_int @ R2 @ K @ ( mult_s3864001451298473021t_unit @ R2 @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_rdistr_int
thf(fact_266_ring_Oadd__pow__rdistr__int,axiom,
! [R2: partia4692342223508353374t_unit,A: nat,B: nat,K: int] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ A @ ( add_po2420080144553951285it_int @ R2 @ K @ B ) )
= ( add_po2420080144553951285it_int @ R2 @ K @ ( mult_n6028127365542633569t_unit @ R2 @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_rdistr_int
thf(fact_267_ring_Oadd__pow__ldistr__int,axiom,
! [R2: partia4934656038542163276t_unit,A: set_int,B: set_int,K: int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( add_po7581009264371422883it_int @ R2 @ K @ A ) @ B )
= ( add_po7581009264371422883it_int @ R2 @ K @ ( mult_s3864001451298473021t_unit @ R2 @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_ldistr_int
thf(fact_268_ring_Oadd__pow__ldistr__int,axiom,
! [R2: partia4692342223508353374t_unit,A: nat,B: nat,K: int] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( add_po2420080144553951285it_int @ R2 @ K @ A ) @ B )
= ( add_po2420080144553951285it_int @ R2 @ K @ ( mult_n6028127365542633569t_unit @ R2 @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_ldistr_int
thf(fact_269_monoid_OUnits__inv__comm,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ X @ Y )
= ( one_na902338870878123981t_unit @ G2 ) )
=> ( ( member_nat @ X @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ Y @ X )
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ) ) ).
% monoid.Units_inv_comm
thf(fact_270_semiring_Or__distr,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ Z @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ Z @ X ) @ ( mult_s3864001451298473021t_unit @ R2 @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_271_semiring_Or__distr,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ Z @ ( add_nat_Product_unit @ R2 @ X @ Y ) )
= ( add_nat_Product_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ Z @ X ) @ ( mult_n6028127365542633569t_unit @ R2 @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_272_semiring_Ol__distr,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Z ) @ ( mult_s3864001451298473021t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_273_semiring_Ol__distr,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( add_nat_Product_unit @ R2 @ X @ Y ) @ Z )
= ( add_nat_Product_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Z ) @ ( mult_n6028127365542633569t_unit @ R2 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_274_semiring_Osemiring__simprules_I9_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( one_se8065767436706823081t_unit @ R2 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_275_semiring_Osemiring__simprules_I9_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( one_na902338870878123981t_unit @ R2 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_276_semiring_Oadd__pow__rdistr,axiom,
! [R2: partia4934656038542163276t_unit,A: set_int,B: set_int,K: nat] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ A @ ( add_po7583499734880473159it_nat @ R2 @ K @ B ) )
= ( add_po7583499734880473159it_nat @ R2 @ K @ ( mult_s3864001451298473021t_unit @ R2 @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_rdistr
thf(fact_277_semiring_Oadd__pow__rdistr,axiom,
! [R2: partia4692342223508353374t_unit,A: nat,B: nat,K: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ A @ ( add_po2422570615063001561it_nat @ R2 @ K @ B ) )
= ( add_po2422570615063001561it_nat @ R2 @ K @ ( mult_n6028127365542633569t_unit @ R2 @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_rdistr
thf(fact_278_semiring_Oadd__pow__ldistr,axiom,
! [R2: partia4934656038542163276t_unit,A: set_int,B: set_int,K: nat] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( add_po7583499734880473159it_nat @ R2 @ K @ A ) @ B )
= ( add_po7583499734880473159it_nat @ R2 @ K @ ( mult_s3864001451298473021t_unit @ R2 @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_ldistr
thf(fact_279_semiring_Oadd__pow__ldistr,axiom,
! [R2: partia4692342223508353374t_unit,A: nat,B: nat,K: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( add_po2422570615063001561it_nat @ R2 @ K @ A ) @ B )
= ( add_po2422570615063001561it_nat @ R2 @ K @ ( mult_n6028127365542633569t_unit @ R2 @ A @ B ) ) ) ) ) ) ).
% semiring.add_pow_ldistr
thf(fact_280_s_Oadd_Onat__pow__Suc2,axiom,
! [X: nat,N: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ ( suc @ N ) @ X )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) ) ) ) ).
% s.add.nat_pow_Suc2
thf(fact_281_ring_Oline__extension__mem__iff,axiom,
! [R2: partia4934656038542163276t_unit,U: set_int,K2: set_set_int,A: set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ U @ ( embedd4283282269743769663t_unit @ R2 @ K2 @ A @ E ) )
= ( ? [X3: set_int] :
( ( member_set_int @ X3 @ K2 )
& ? [Y4: set_int] :
( ( member_set_int @ Y4 @ E )
& ( U
= ( add_se5859248395121729892t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ X3 @ A ) @ Y4 ) ) ) ) ) ) ) ).
% ring.line_extension_mem_iff
thf(fact_282_ring_Oline__extension__mem__iff,axiom,
! [R2: partia4692342223508353374t_unit,U: nat,K2: set_nat,A: nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ U @ ( embedd838748496991043025t_unit @ R2 @ K2 @ A @ E ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ K2 )
& ? [Y4: nat] :
( ( member_nat @ Y4 @ E )
& ( U
= ( add_nat_Product_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ X3 @ A ) @ Y4 ) ) ) ) ) ) ) ).
% ring.line_extension_mem_iff
thf(fact_283_s_Oline__extension__in__carrier,axiom,
! [K2: set_nat,A: nat,E: set_nat] :
( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ E @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ord_less_eq_set_nat @ ( embedd838748496991043025t_unit @ ( mod_ring @ n ) @ K2 @ A @ E ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ) ).
% s.line_extension_in_carrier
thf(fact_284_s_Ogroup__l__invI,axiom,
( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
& ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Xa @ X4 )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) )
=> ( group_2402336746480010918t_unit @ ( mod_ring @ n ) ) ) ).
% s.group_l_invI
thf(fact_285_s_Oadd_Oint__pow__pow,axiom,
! [X: nat,M: int,N: int] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ M @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ N @ X ) )
= ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ ( times_times_int @ N @ M ) @ X ) ) ) ).
% s.add.int_pow_pow
thf(fact_286_s_Oadd_Onat__pow__pow,axiom,
! [X: nat,M: nat,N: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ M @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) )
= ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ ( times_times_nat @ N @ M ) @ X ) ) ) ).
% s.add.nat_pow_pow
thf(fact_287_s_Oadd_Oinv__comm,axiom,
! [X: nat,Y: nat] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ) ).
% s.add.inv_comm
thf(fact_288_s_Oadd_Ol__inv__ex,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
& ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X4 @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.add.l_inv_ex
thf(fact_289_s_Oadd_Oone__unique,axiom,
! [U: nat] :
( ( member_nat @ U @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ U @ X4 )
= X4 ) )
=> ( U
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.add.one_unique
thf(fact_290_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_291_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_292_s_Ominus__unique,axiom,
! [Y: nat,X: nat,Y2: nat] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y2 )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% s.minus_unique
thf(fact_293_s_Oadd_Or__inv__ex,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
& ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ X4 )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.add.r_inv_ex
thf(fact_294_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_295_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_296_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_297_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_298_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_299_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_300_s_Ozero__closed,axiom,
member_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ).
% s.zero_closed
thf(fact_301_s_Oadd_Onat__pow__one,axiom,
! [N: nat] :
( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ).
% s.add.nat_pow_one
thf(fact_302_s_Oadd_Oint__pow__one,axiom,
! [Z: int] :
( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ Z @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ).
% s.add.int_pow_one
thf(fact_303_s_Or__zero,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
= X ) ) ).
% s.r_zero
thf(fact_304_s_Ol__zero,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ X )
= X ) ) ).
% s.l_zero
thf(fact_305_s_Oadd_Or__cancel__one_H,axiom,
! [X: nat,A: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( X
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ A @ X ) )
= ( A
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ) ).
% s.add.r_cancel_one'
thf(fact_306_s_Oadd_Or__cancel__one,axiom,
! [X: nat,A: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ A @ X )
= X )
= ( A
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ) ).
% s.add.r_cancel_one
thf(fact_307_s_Oadd_Ol__cancel__one_H,axiom,
! [X: nat,A: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( X
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ A ) )
= ( A
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ) ).
% s.add.l_cancel_one'
thf(fact_308_s_Oadd_Ol__cancel__one,axiom,
! [X: nat,A: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ A )
= X )
= ( A
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ) ).
% s.add.l_cancel_one
thf(fact_309_s_Or__null,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.r_null
thf(fact_310_s_Ol__null,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.l_null
thf(fact_311_s_Oadd_Onat__pow__Suc,axiom,
! [N: nat,X: nat] :
( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ ( suc @ N ) @ X )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) @ X ) ) ).
% s.add.nat_pow_Suc
thf(fact_312_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_313_group_Ois__group,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( group_2402336746480010918t_unit @ G2 ) ) ).
% group.is_group
thf(fact_314_group_Ois__group,axiom,
! [G2: partia6999121085262744519t_unit] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( group_1007331108230521615t_unit @ G2 ) ) ).
% group.is_group
thf(fact_315_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_316_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_317_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_318_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_319_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_320_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_321_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_322_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_323_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_324_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_325_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_326_group_OUnits,axiom,
! [G2: partia6999121085262744519t_unit] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ord_less_eq_set_nat @ ( partia5989097197943085638t_unit @ G2 ) @ ( units_1133518744711867743t_unit @ G2 ) ) ) ).
% group.Units
thf(fact_327_group_OUnits,axiom,
! [G2: partia4934656038542163276t_unit] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ord_le4403425263959731960et_int @ ( partia966996272515721803t_unit @ G2 ) @ ( units_4038138251425117394t_unit @ G2 ) ) ) ).
% group.Units
thf(fact_328_group_OUnits,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ord_less_eq_set_nat @ ( partia3499330772048238685t_unit @ G2 ) @ ( units_1295200668666280182t_unit @ G2 ) ) ) ).
% group.Units
thf(fact_329_group_Ois__monoid,axiom,
! [G2: partia6999121085262744519t_unit] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( monoid5435825805542372298t_unit @ G2 ) ) ).
% group.is_monoid
thf(fact_330_group_Ois__monoid,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( monoid4477821099152670945t_unit @ G2 ) ) ).
% group.is_monoid
thf(fact_331_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_332_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_333_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_334_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_335_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_336_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_337_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_338_Group_Ogroup_Oright__cancel,axiom,
! [G2: partia6999121085262744519t_unit,X: nat,Y: nat,Z: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ Z @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( ( mult_n7532461433716462410t_unit @ G2 @ Y @ X )
= ( mult_n7532461433716462410t_unit @ G2 @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% Group.group.right_cancel
thf(fact_339_Group_Ogroup_Oright__cancel,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( ( mult_s3864001451298473021t_unit @ G2 @ Y @ X )
= ( mult_s3864001451298473021t_unit @ G2 @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% Group.group.right_cancel
thf(fact_340_Group_Ogroup_Oright__cancel,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ Y @ X )
= ( mult_n6028127365542633569t_unit @ G2 @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ) ).
% Group.group.right_cancel
thf(fact_341_group_OUnits__eq,axiom,
! [G2: partia6999121085262744519t_unit] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( units_1133518744711867743t_unit @ G2 )
= ( partia5989097197943085638t_unit @ G2 ) ) ) ).
% group.Units_eq
thf(fact_342_group_OUnits__eq,axiom,
! [G2: partia4934656038542163276t_unit] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( units_4038138251425117394t_unit @ G2 )
= ( partia966996272515721803t_unit @ G2 ) ) ) ).
% group.Units_eq
thf(fact_343_group_OUnits__eq,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( units_1295200668666280182t_unit @ G2 )
= ( partia3499330772048238685t_unit @ G2 ) ) ) ).
% group.Units_eq
thf(fact_344_ring_Oring__simprules_I2_J,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% ring.ring_simprules(2)
thf(fact_345_ring_Oring__simprules_I2_J,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% ring.ring_simprules(2)
thf(fact_346_abelian__groupE_I2_J,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( abelia23968383328945916t_unit @ R2 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% abelian_groupE(2)
thf(fact_347_abelian__groupE_I2_J,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( abelia406319425121669262t_unit @ R2 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% abelian_groupE(2)
thf(fact_348_abelian__monoidE_I2_J,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( abelia3815030880812984441t_unit @ R2 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% abelian_monoidE(2)
thf(fact_349_abelian__monoidE_I2_J,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( abelia362511065248671243t_unit @ R2 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% abelian_monoidE(2)
thf(fact_350_abelian__monoid_Ozero__closed,axiom,
! [G2: partia4934656038542163276t_unit] :
( ( abelia3815030880812984441t_unit @ G2 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ G2 ) @ ( partia966996272515721803t_unit @ G2 ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_351_abelian__monoid_Ozero__closed,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( abelia362511065248671243t_unit @ G2 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ G2 ) @ ( partia3499330772048238685t_unit @ G2 ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_352_ring__hom__zero,axiom,
! [H: set_int > set_int,R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit] :
( ( member5205197933313416826et_int @ H @ ( ring_h3404898052528352314t_unit @ R2 @ S ) )
=> ( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ring_s5316885176909347197t_unit @ S )
=> ( ( H @ ( zero_s6269048424454532197t_unit @ R2 ) )
= ( zero_s6269048424454532197t_unit @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_353_ring__hom__zero,axiom,
! [H: set_int > nat,R2: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit] :
( ( member_set_int_nat @ H @ ( ring_h9051218956466089420t_unit @ R2 @ S ) )
=> ( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ring_n9194430563101542159t_unit @ S )
=> ( ( H @ ( zero_s6269048424454532197t_unit @ R2 ) )
= ( zero_n5149899317435570679t_unit @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_354_ring__hom__zero,axiom,
! [H: nat > set_int,R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit] :
( ( member_nat_set_int @ H @ ( ring_h4752909569380436264t_unit @ R2 @ S ) )
=> ( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ring_s5316885176909347197t_unit @ S )
=> ( ( H @ ( zero_n5149899317435570679t_unit @ R2 ) )
= ( zero_s6269048424454532197t_unit @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_355_ring__hom__zero,axiom,
! [H: nat > nat,R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit] :
( ( member_nat_nat @ H @ ( ring_h4412161176302437050t_unit @ R2 @ S ) )
=> ( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ring_n9194430563101542159t_unit @ S )
=> ( ( H @ ( zero_n5149899317435570679t_unit @ R2 ) )
= ( zero_n5149899317435570679t_unit @ S ) ) ) ) ) ).
% ring_hom_zero
thf(fact_356_semiring_Osemiring__simprules_I2_J,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_357_semiring_Osemiring__simprules_I2_J,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_358_ring_Oline__extension__in__carrier,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,A: set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ord_le4403425263959731960et_int @ K2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ord_le4403425263959731960et_int @ ( embedd4283282269743769663t_unit @ R2 @ K2 @ A @ E ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_359_ring_Oline__extension__in__carrier,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,A: nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( ord_less_eq_set_nat @ E @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ord_less_eq_set_nat @ ( embedd838748496991043025t_unit @ R2 @ K2 @ A @ E ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_360_groupI,axiom,
! [G2: partia6999121085262744519t_unit] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia5989097197943085638t_unit @ G2 ) )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( member_nat @ ( mult_n7532461433716462410t_unit @ G2 @ X4 @ Y3 ) @ ( partia5989097197943085638t_unit @ G2 ) ) ) )
=> ( ( member_nat @ ( one_nat_Product_unit @ G2 ) @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia5989097197943085638t_unit @ G2 ) )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( partia5989097197943085638t_unit @ G2 ) )
=> ! [Z2: nat] :
( ( member_nat @ Z2 @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( mult_n7532461433716462410t_unit @ G2 @ ( mult_n7532461433716462410t_unit @ G2 @ X4 @ Y3 ) @ Z2 )
= ( mult_n7532461433716462410t_unit @ G2 @ X4 @ ( mult_n7532461433716462410t_unit @ G2 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( mult_n7532461433716462410t_unit @ G2 @ ( one_nat_Product_unit @ G2 ) @ X4 )
= X4 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia5989097197943085638t_unit @ G2 ) )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ ( partia5989097197943085638t_unit @ G2 ) )
& ( ( mult_n7532461433716462410t_unit @ G2 @ Xa @ X4 )
= ( one_nat_Product_unit @ G2 ) ) ) )
=> ( group_1007331108230521615t_unit @ G2 ) ) ) ) ) ) ).
% groupI
thf(fact_361_groupI,axiom,
! [G2: partia4934656038542163276t_unit] :
( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
=> ! [Y3: set_int] :
( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ G2 @ X4 @ Y3 ) @ ( partia966996272515721803t_unit @ G2 ) ) ) )
=> ( ( member_set_int @ ( one_se8065767436706823081t_unit @ G2 ) @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
=> ! [Y3: set_int] :
( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ G2 ) )
=> ! [Z2: set_int] :
( ( member_set_int @ Z2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( mult_s3864001451298473021t_unit @ G2 @ ( mult_s3864001451298473021t_unit @ G2 @ X4 @ Y3 ) @ Z2 )
= ( mult_s3864001451298473021t_unit @ G2 @ X4 @ ( mult_s3864001451298473021t_unit @ G2 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( mult_s3864001451298473021t_unit @ G2 @ ( one_se8065767436706823081t_unit @ G2 ) @ X4 )
= X4 ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
=> ? [Xa: set_int] :
( ( member_set_int @ Xa @ ( partia966996272515721803t_unit @ G2 ) )
& ( ( mult_s3864001451298473021t_unit @ G2 @ Xa @ X4 )
= ( one_se8065767436706823081t_unit @ G2 ) ) ) )
=> ( group_1453825718996320898t_unit @ G2 ) ) ) ) ) ) ).
% groupI
thf(fact_362_groupI,axiom,
! [G2: partia4692342223508353374t_unit] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ G2 @ X4 @ Y3 ) @ ( partia3499330772048238685t_unit @ G2 ) ) ) )
=> ( ( member_nat @ ( one_na902338870878123981t_unit @ G2 ) @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ! [Z2: nat] :
( ( member_nat @ Z2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ ( mult_n6028127365542633569t_unit @ G2 @ X4 @ Y3 ) @ Z2 )
= ( mult_n6028127365542633569t_unit @ G2 @ X4 @ ( mult_n6028127365542633569t_unit @ G2 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ ( one_na902338870878123981t_unit @ G2 ) @ X4 )
= X4 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ ( partia3499330772048238685t_unit @ G2 ) )
& ( ( mult_n6028127365542633569t_unit @ G2 @ Xa @ X4 )
= ( one_na902338870878123981t_unit @ G2 ) ) ) )
=> ( group_2402336746480010918t_unit @ G2 ) ) ) ) ) ) ).
% groupI
thf(fact_363_group_Or__cancel__one_H,axiom,
! [G2: partia6999121085262744519t_unit,X: nat,A: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ A @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( X
= ( mult_n7532461433716462410t_unit @ G2 @ A @ X ) )
= ( A
= ( one_nat_Product_unit @ G2 ) ) ) ) ) ) ).
% group.r_cancel_one'
thf(fact_364_group_Or__cancel__one_H,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,A: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( X
= ( mult_s3864001451298473021t_unit @ G2 @ A @ X ) )
= ( A
= ( one_se8065767436706823081t_unit @ G2 ) ) ) ) ) ) ).
% group.r_cancel_one'
thf(fact_365_group_Or__cancel__one_H,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,A: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( X
= ( mult_n6028127365542633569t_unit @ G2 @ A @ X ) )
= ( A
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ) ) ).
% group.r_cancel_one'
thf(fact_366_group_Ol__cancel__one_H,axiom,
! [G2: partia6999121085262744519t_unit,X: nat,A: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ A @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( X
= ( mult_n7532461433716462410t_unit @ G2 @ X @ A ) )
= ( A
= ( one_nat_Product_unit @ G2 ) ) ) ) ) ) ).
% group.l_cancel_one'
thf(fact_367_group_Ol__cancel__one_H,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,A: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( X
= ( mult_s3864001451298473021t_unit @ G2 @ X @ A ) )
= ( A
= ( one_se8065767436706823081t_unit @ G2 ) ) ) ) ) ) ).
% group.l_cancel_one'
thf(fact_368_group_Ol__cancel__one_H,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,A: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( X
= ( mult_n6028127365542633569t_unit @ G2 @ X @ A ) )
= ( A
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ) ) ).
% group.l_cancel_one'
thf(fact_369_group_Or__cancel__one,axiom,
! [G2: partia6999121085262744519t_unit,X: nat,A: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ A @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( ( mult_n7532461433716462410t_unit @ G2 @ A @ X )
= X )
= ( A
= ( one_nat_Product_unit @ G2 ) ) ) ) ) ) ).
% group.r_cancel_one
thf(fact_370_group_Or__cancel__one,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,A: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( ( mult_s3864001451298473021t_unit @ G2 @ A @ X )
= X )
= ( A
= ( one_se8065767436706823081t_unit @ G2 ) ) ) ) ) ) ).
% group.r_cancel_one
thf(fact_371_group_Or__cancel__one,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,A: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ A @ X )
= X )
= ( A
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ) ) ).
% group.r_cancel_one
thf(fact_372_group_Ol__cancel__one,axiom,
! [G2: partia6999121085262744519t_unit,X: nat,A: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ A @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( ( mult_n7532461433716462410t_unit @ G2 @ X @ A )
= X )
= ( A
= ( one_nat_Product_unit @ G2 ) ) ) ) ) ) ).
% group.l_cancel_one
thf(fact_373_group_Ol__cancel__one,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,A: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( ( mult_s3864001451298473021t_unit @ G2 @ X @ A )
= X )
= ( A
= ( one_se8065767436706823081t_unit @ G2 ) ) ) ) ) ) ).
% group.l_cancel_one
thf(fact_374_group_Ol__cancel__one,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,A: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ X @ A )
= X )
= ( A
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ) ) ).
% group.l_cancel_one
thf(fact_375_group_Or__inv__ex,axiom,
! [G2: partia6999121085262744519t_unit,X: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( partia5989097197943085638t_unit @ G2 ) )
& ( ( mult_n7532461433716462410t_unit @ G2 @ X @ X4 )
= ( one_nat_Product_unit @ G2 ) ) ) ) ) ).
% group.r_inv_ex
thf(fact_376_group_Or__inv__ex,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
& ( ( mult_s3864001451298473021t_unit @ G2 @ X @ X4 )
= ( one_se8065767436706823081t_unit @ G2 ) ) ) ) ) ).
% group.r_inv_ex
thf(fact_377_group_Or__inv__ex,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
& ( ( mult_n6028127365542633569t_unit @ G2 @ X @ X4 )
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ) ).
% group.r_inv_ex
thf(fact_378_group_Ol__inv__ex,axiom,
! [G2: partia6999121085262744519t_unit,X: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( partia5989097197943085638t_unit @ G2 ) )
& ( ( mult_n7532461433716462410t_unit @ G2 @ X4 @ X )
= ( one_nat_Product_unit @ G2 ) ) ) ) ) ).
% group.l_inv_ex
thf(fact_379_group_Ol__inv__ex,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
& ( ( mult_s3864001451298473021t_unit @ G2 @ X4 @ X )
= ( one_se8065767436706823081t_unit @ G2 ) ) ) ) ) ).
% group.l_inv_ex
thf(fact_380_group_Ol__inv__ex,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
& ( ( mult_n6028127365542633569t_unit @ G2 @ X4 @ X )
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ) ).
% group.l_inv_ex
thf(fact_381_group_Oinv__comm,axiom,
! [G2: partia6999121085262744519t_unit,X: nat,Y: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( ( mult_n7532461433716462410t_unit @ G2 @ X @ Y )
= ( one_nat_Product_unit @ G2 ) )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( mult_n7532461433716462410t_unit @ G2 @ Y @ X )
= ( one_nat_Product_unit @ G2 ) ) ) ) ) ) ).
% group.inv_comm
thf(fact_382_group_Oinv__comm,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( ( mult_s3864001451298473021t_unit @ G2 @ X @ Y )
= ( one_se8065767436706823081t_unit @ G2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( mult_s3864001451298473021t_unit @ G2 @ Y @ X )
= ( one_se8065767436706823081t_unit @ G2 ) ) ) ) ) ) ).
% group.inv_comm
thf(fact_383_group_Oinv__comm,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ X @ Y )
= ( one_na902338870878123981t_unit @ G2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ Y @ X )
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ) ) ).
% group.inv_comm
thf(fact_384_ring_Oring__simprules_I15_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ ( zero_s6269048424454532197t_unit @ R2 ) )
= X ) ) ) ).
% ring.ring_simprules(15)
thf(fact_385_ring_Oring__simprules_I15_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ ( zero_n5149899317435570679t_unit @ R2 ) )
= X ) ) ) ).
% ring.ring_simprules(15)
thf(fact_386_ring_Oring__simprules_I8_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( zero_s6269048424454532197t_unit @ R2 ) @ X )
= X ) ) ) ).
% ring.ring_simprules(8)
thf(fact_387_ring_Oring__simprules_I8_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( zero_n5149899317435570679t_unit @ R2 ) @ X )
= X ) ) ) ).
% ring.ring_simprules(8)
thf(fact_388_ring_Oring__simprules_I25_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ X @ ( zero_s6269048424454532197t_unit @ R2 ) )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_389_ring_Oring__simprules_I25_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ X @ ( zero_n5149899317435570679t_unit @ R2 ) )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_390_ring_Oring__simprules_I24_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( zero_s6269048424454532197t_unit @ R2 ) @ X )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_391_ring_Oring__simprules_I24_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( zero_n5149899317435570679t_unit @ R2 ) @ X )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_392_abelian__groupE_I6_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( abelia23968383328945916t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
& ( ( add_se5859248395121729892t_unit @ R2 @ X4 @ X )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_393_abelian__groupE_I6_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( abelia406319425121669262t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
& ( ( add_nat_Product_unit @ R2 @ X4 @ X )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_394_abelian__groupE_I5_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( abelia23968383328945916t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( zero_s6269048424454532197t_unit @ R2 ) @ X )
= X ) ) ) ).
% abelian_groupE(5)
thf(fact_395_abelian__groupE_I5_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( abelia406319425121669262t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( zero_n5149899317435570679t_unit @ R2 ) @ X )
= X ) ) ) ).
% abelian_groupE(5)
thf(fact_396_abelian__groupI,axiom,
! [R2: partia4934656038542163276t_unit] :
( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ! [Y3: set_int] :
( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R2 @ X4 @ Y3 ) @ ( partia966996272515721803t_unit @ R2 ) ) ) )
=> ( ( member_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ! [Y3: set_int] :
( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R2 ) )
=> ! [Z2: set_int] :
( ( member_set_int @ Z2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( add_se5859248395121729892t_unit @ R2 @ X4 @ Y3 ) @ Z2 )
= ( add_se5859248395121729892t_unit @ R2 @ X4 @ ( add_se5859248395121729892t_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ! [Y3: set_int] :
( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X4 @ Y3 )
= ( add_se5859248395121729892t_unit @ R2 @ Y3 @ X4 ) ) ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( zero_s6269048424454532197t_unit @ R2 ) @ X4 )
= X4 ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ? [Xa: set_int] :
( ( member_set_int @ Xa @ ( partia966996272515721803t_unit @ R2 ) )
& ( ( add_se5859248395121729892t_unit @ R2 @ Xa @ X4 )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) )
=> ( abelia23968383328945916t_unit @ R2 ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_397_abelian__groupI,axiom,
! [R2: partia4692342223508353374t_unit] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ X4 @ Y3 ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) )
=> ( ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ! [Z2: nat] :
( ( member_nat @ Z2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( add_nat_Product_unit @ R2 @ X4 @ Y3 ) @ Z2 )
= ( add_nat_Product_unit @ R2 @ X4 @ ( add_nat_Product_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X4 @ Y3 )
= ( add_nat_Product_unit @ R2 @ Y3 @ X4 ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( zero_n5149899317435570679t_unit @ R2 ) @ X4 )
= X4 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ ( partia3499330772048238685t_unit @ R2 ) )
& ( ( add_nat_Product_unit @ R2 @ Xa @ X4 )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) )
=> ( abelia406319425121669262t_unit @ R2 ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_398_abelian__monoidE_I4_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( abelia3815030880812984441t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( zero_s6269048424454532197t_unit @ R2 ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_399_abelian__monoidE_I4_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( abelia362511065248671243t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( zero_n5149899317435570679t_unit @ R2 ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_400_abelian__monoid_Ol__zero,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( abelia3815030880812984441t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( add_se5859248395121729892t_unit @ G2 @ ( zero_s6269048424454532197t_unit @ G2 ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_401_abelian__monoid_Ol__zero,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( abelia362511065248671243t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( add_nat_Product_unit @ G2 @ ( zero_n5149899317435570679t_unit @ G2 ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_402_abelian__monoid_Or__zero,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( abelia3815030880812984441t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( add_se5859248395121729892t_unit @ G2 @ X @ ( zero_s6269048424454532197t_unit @ G2 ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_403_abelian__monoid_Or__zero,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( abelia362511065248671243t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( add_nat_Product_unit @ G2 @ X @ ( zero_n5149899317435570679t_unit @ G2 ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_404_abelian__monoid_Ominus__unique,axiom,
! [G2: partia4934656038542163276t_unit,Y: set_int,X: set_int,Y2: set_int] :
( ( abelia3815030880812984441t_unit @ G2 )
=> ( ( ( add_se5859248395121729892t_unit @ G2 @ Y @ X )
= ( zero_s6269048424454532197t_unit @ G2 ) )
=> ( ( ( add_se5859248395121729892t_unit @ G2 @ X @ Y2 )
= ( zero_s6269048424454532197t_unit @ G2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_405_abelian__monoid_Ominus__unique,axiom,
! [G2: partia4692342223508353374t_unit,Y: nat,X: nat,Y2: nat] :
( ( abelia362511065248671243t_unit @ G2 )
=> ( ( ( add_nat_Product_unit @ G2 @ Y @ X )
= ( zero_n5149899317435570679t_unit @ G2 ) )
=> ( ( ( add_nat_Product_unit @ G2 @ X @ Y2 )
= ( zero_n5149899317435570679t_unit @ G2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_406_abelian__monoidI,axiom,
! [R2: partia4934656038542163276t_unit] :
( ! [X4: set_int,Y3: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R2 @ X4 @ Y3 ) @ ( partia966996272515721803t_unit @ R2 ) ) ) )
=> ( ( member_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ! [X4: set_int,Y3: set_int,Z2: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Z2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( add_se5859248395121729892t_unit @ R2 @ X4 @ Y3 ) @ Z2 )
= ( add_se5859248395121729892t_unit @ R2 @ X4 @ ( add_se5859248395121729892t_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( zero_s6269048424454532197t_unit @ R2 ) @ X4 )
= X4 ) )
=> ( ! [X4: set_int,Y3: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X4 @ Y3 )
= ( add_se5859248395121729892t_unit @ R2 @ Y3 @ X4 ) ) ) )
=> ( abelia3815030880812984441t_unit @ R2 ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_407_abelian__monoidI,axiom,
! [R2: partia4692342223508353374t_unit] :
( ! [X4: nat,Y3: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ X4 @ Y3 ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) )
=> ( ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ! [X4: nat,Y3: nat,Z2: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Z2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( add_nat_Product_unit @ R2 @ X4 @ Y3 ) @ Z2 )
= ( add_nat_Product_unit @ R2 @ X4 @ ( add_nat_Product_unit @ R2 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( zero_n5149899317435570679t_unit @ R2 ) @ X4 )
= X4 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X4 @ Y3 )
= ( add_nat_Product_unit @ R2 @ Y3 @ X4 ) ) ) )
=> ( abelia362511065248671243t_unit @ R2 ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_408_semiring_Osemiring__simprules_I11_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ ( zero_s6269048424454532197t_unit @ R2 ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_409_semiring_Osemiring__simprules_I11_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ ( zero_n5149899317435570679t_unit @ R2 ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_410_semiring_Osemiring__simprules_I6_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( zero_s6269048424454532197t_unit @ R2 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_411_semiring_Osemiring__simprules_I6_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( zero_n5149899317435570679t_unit @ R2 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_412_semiring_Ol__null,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( zero_s6269048424454532197t_unit @ R2 ) @ X )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% semiring.l_null
thf(fact_413_semiring_Ol__null,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( zero_n5149899317435570679t_unit @ R2 ) @ X )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% semiring.l_null
thf(fact_414_semiring_Or__null,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ X @ ( zero_s6269048424454532197t_unit @ R2 ) )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% semiring.r_null
thf(fact_415_semiring_Or__null,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ X @ ( zero_n5149899317435570679t_unit @ R2 ) )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% semiring.r_null
thf(fact_416_ring_Oline__extension_Ocong,axiom,
embedd838748496991043025t_unit = embedd838748496991043025t_unit ).
% ring.line_extension.cong
thf(fact_417_monoid_Ogroup__l__invI,axiom,
! [G2: partia6999121085262744519t_unit] :
( ( monoid5435825805542372298t_unit @ G2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia5989097197943085638t_unit @ G2 ) )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ ( partia5989097197943085638t_unit @ G2 ) )
& ( ( mult_n7532461433716462410t_unit @ G2 @ Xa @ X4 )
= ( one_nat_Product_unit @ G2 ) ) ) )
=> ( group_1007331108230521615t_unit @ G2 ) ) ) ).
% monoid.group_l_invI
thf(fact_418_monoid_Ogroup__l__invI,axiom,
! [G2: partia4934656038542163276t_unit] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G2 ) )
=> ? [Xa: set_int] :
( ( member_set_int @ Xa @ ( partia966996272515721803t_unit @ G2 ) )
& ( ( mult_s3864001451298473021t_unit @ G2 @ Xa @ X4 )
= ( one_se8065767436706823081t_unit @ G2 ) ) ) )
=> ( group_1453825718996320898t_unit @ G2 ) ) ) ).
% monoid.group_l_invI
thf(fact_419_monoid_Ogroup__l__invI,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ ( partia3499330772048238685t_unit @ G2 ) )
& ( ( mult_n6028127365542633569t_unit @ G2 @ Xa @ X4 )
= ( one_na902338870878123981t_unit @ G2 ) ) ) )
=> ( group_2402336746480010918t_unit @ G2 ) ) ) ).
% monoid.group_l_invI
thf(fact_420_s_Oa__lcos__mult__one,axiom,
! [M2: set_nat] :
( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_l_co3340896127261556338t_unit @ ( mod_ring @ n ) @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ M2 )
= M2 ) ) ).
% s.a_lcos_mult_one
thf(fact_421_s_Oa__lcos__m__assoc,axiom,
! [M2: set_nat,G: nat,H: nat] :
( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ G @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ H @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_l_co3340896127261556338t_unit @ ( mod_ring @ n ) @ G @ ( a_l_co3340896127261556338t_unit @ ( mod_ring @ n ) @ H @ M2 ) )
= ( a_l_co3340896127261556338t_unit @ ( mod_ring @ n ) @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ G @ H ) @ M2 ) ) ) ) ) ).
% s.a_lcos_m_assoc
thf(fact_422_s_Osubalgebra__in__carrier,axiom,
! [K2: set_nat,V: set_nat] :
( ( embedd2609395410403458802t_unit @ K2 @ V @ ( mod_ring @ n ) )
=> ( ord_less_eq_set_nat @ V @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.subalgebra_in_carrier
thf(fact_423_s_Ocarrier__is__subalgebra,axiom,
! [K2: set_nat] :
( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( embedd2609395410403458802t_unit @ K2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) @ ( mod_ring @ n ) ) ) ).
% s.carrier_is_subalgebra
thf(fact_424_s_Ozeropideal,axiom,
princi4652470909602072491t_unit @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) @ ( mod_ring @ n ) ).
% s.zeropideal
thf(fact_425_s_Oa__l__coset__subset__G,axiom,
! [H2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ord_less_eq_set_nat @ ( a_l_co3340896127261556338t_unit @ ( mod_ring @ n ) @ X @ H2 ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.a_l_coset_subset_G
thf(fact_426_s_Oone__zeroI,axiom,
( ( ( partia3499330772048238685t_unit @ ( mod_ring @ n ) )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) )
=> ( ( one_na902338870878123981t_unit @ ( mod_ring @ n ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.one_zeroI
thf(fact_427_s_Oone__zeroD,axiom,
( ( ( one_na902338870878123981t_unit @ ( mod_ring @ n ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
=> ( ( partia3499330772048238685t_unit @ ( mod_ring @ n ) )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) ) ).
% s.one_zeroD
thf(fact_428_s_Ocarrier__one__not__zero,axiom,
( ( ( partia3499330772048238685t_unit @ ( mod_ring @ n ) )
!= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) )
= ( ( one_na902338870878123981t_unit @ ( mod_ring @ n ) )
!= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.carrier_one_not_zero
thf(fact_429_s_Ocarrier__one__zero,axiom,
( ( ( partia3499330772048238685t_unit @ ( mod_ring @ n ) )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) )
= ( ( one_na902338870878123981t_unit @ ( mod_ring @ n ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.carrier_one_zero
thf(fact_430_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_431_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_432_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_433_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_434_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_435_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_436_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_437_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_438_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_439_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_440_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_441_Suc__le__D,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M3 )
=> ? [M4: nat] :
( M3
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_442_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_443_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_444_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_445_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_446_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_447_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R2 @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z2: nat] :
( ( R2 @ X4 @ Y3 )
=> ( ( R2 @ Y3 @ Z2 )
=> ( R2 @ X4 @ Z2 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_448_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_449_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_450_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_451_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_452_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_453_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_454_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_455_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_456_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_457_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_458_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_459_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_460_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_461_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_462_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_463_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_464_subalgebra_Osmult__closed,axiom,
! [K2: set_nat,V: set_nat,R2: partia4692342223508353374t_unit,K: nat,V2: nat] :
( ( embedd2609395410403458802t_unit @ K2 @ V @ R2 )
=> ( ( member_nat @ K @ K2 )
=> ( ( member_nat @ V2 @ V )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ K @ V2 ) @ V ) ) ) ) ).
% subalgebra.smult_closed
thf(fact_465_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_466_ring_Osubalgebra__in__carrier,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,V: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( embedd2743979684206749024t_unit @ K2 @ V @ R2 )
=> ( ord_le4403425263959731960et_int @ V @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_467_ring_Osubalgebra__in__carrier,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,V: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( embedd2609395410403458802t_unit @ K2 @ V @ R2 )
=> ( ord_less_eq_set_nat @ V @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_468_ring_Ocarrier__is__subalgebra,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ord_le4403425263959731960et_int @ K2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( embedd2743979684206749024t_unit @ K2 @ ( partia966996272515721803t_unit @ R2 ) @ R2 ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_469_ring_Ocarrier__is__subalgebra,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( embedd2609395410403458802t_unit @ K2 @ ( partia3499330772048238685t_unit @ R2 ) @ R2 ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_470_ring_Ozeropideal,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( princi8860937869964495385t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) @ R2 ) ) ).
% ring.zeropideal
thf(fact_471_ring_Ozeropideal,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( princi4652470909602072491t_unit @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) @ R2 ) ) ).
% ring.zeropideal
thf(fact_472_semiring_Ocarrier__one__not__zero,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( ( partia966996272515721803t_unit @ R2 )
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) )
= ( ( one_se8065767436706823081t_unit @ R2 )
!= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_473_semiring_Ocarrier__one__not__zero,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( ( partia3499330772048238685t_unit @ R2 )
!= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) )
= ( ( one_na902338870878123981t_unit @ R2 )
!= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_474_semiring_Ocarrier__one__zero,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( ( partia966996272515721803t_unit @ R2 )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) )
= ( ( one_se8065767436706823081t_unit @ R2 )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_475_semiring_Ocarrier__one__zero,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( ( partia3499330772048238685t_unit @ R2 )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) )
= ( ( one_na902338870878123981t_unit @ R2 )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_476_semiring_Oone__zeroI,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( ( partia966996272515721803t_unit @ R2 )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) )
=> ( ( one_se8065767436706823081t_unit @ R2 )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% semiring.one_zeroI
thf(fact_477_semiring_Oone__zeroI,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( ( partia3499330772048238685t_unit @ R2 )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) )
=> ( ( one_na902338870878123981t_unit @ R2 )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% semiring.one_zeroI
thf(fact_478_semiring_Oone__zeroD,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( ( ( one_se8065767436706823081t_unit @ R2 )
= ( zero_s6269048424454532197t_unit @ R2 ) )
=> ( ( partia966996272515721803t_unit @ R2 )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) ) ) ).
% semiring.one_zeroD
thf(fact_479_semiring_Oone__zeroD,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R2 )
=> ( ( ( one_na902338870878123981t_unit @ R2 )
= ( zero_n5149899317435570679t_unit @ R2 ) )
=> ( ( partia3499330772048238685t_unit @ R2 )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) ) ) ).
% semiring.one_zeroD
thf(fact_480_s_Ounits__group,axiom,
group_1007331108230521615t_unit @ ( units_2222427131560811679t_unit @ ( mod_ring @ n ) ) ).
% s.units_group
thf(fact_481_s_Ogenideal__one,axiom,
( ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ ( insert_nat @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) )
= ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ).
% s.genideal_one
thf(fact_482_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A3: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_483_singleton__insert__inj__eq_H,axiom,
! [A: nat,A3: set_nat,B: nat] :
( ( ( insert_nat @ A @ A3 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_484_s_OIdl__subset__ideal_H,axiom,
! [A: nat,B: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( member_nat @ A @ ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ) ).
% s.Idl_subset_ideal'
thf(fact_485_s_Ogenideal__zero,axiom,
( ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) ).
% s.genideal_zero
thf(fact_486_s_Ogenideal__self_H,axiom,
! [I: nat] :
( ( member_nat @ I @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ I @ ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) ) ) ).
% s.genideal_self'
thf(fact_487_subsetI,axiom,
! [A3: set_nat_set_int,B5: set_nat_set_int] :
( ! [X4: nat > set_int] :
( ( member_nat_set_int @ X4 @ A3 )
=> ( member_nat_set_int @ X4 @ B5 ) )
=> ( ord_le5995675665013768039et_int @ A3 @ B5 ) ) ).
% subsetI
thf(fact_488_subsetI,axiom,
! [A3: set_set_nat,B5: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A3 )
=> ( member_set_nat @ X4 @ B5 ) )
=> ( ord_le6893508408891458716et_nat @ A3 @ B5 ) ) ).
% subsetI
thf(fact_489_subsetI,axiom,
! [A3: set_nat,B5: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_nat @ X4 @ B5 ) )
=> ( ord_less_eq_set_nat @ A3 @ B5 ) ) ).
% subsetI
thf(fact_490_subset__antisym,axiom,
! [A3: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ A3 )
=> ( A3 = B5 ) ) ) ).
% subset_antisym
thf(fact_491_empty__iff,axiom,
! [C: nat > set_int] :
~ ( member_nat_set_int @ C @ bot_bo8417611410066262939et_int ) ).
% empty_iff
thf(fact_492_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_493_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_494_all__not__in__conv,axiom,
! [A3: set_nat_set_int] :
( ( ! [X3: nat > set_int] :
~ ( member_nat_set_int @ X3 @ A3 ) )
= ( A3 = bot_bo8417611410066262939et_int ) ) ).
% all_not_in_conv
thf(fact_495_all__not__in__conv,axiom,
! [A3: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A3 ) )
= ( A3 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_496_all__not__in__conv,axiom,
! [A3: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_497_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_498_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_499_s_Osubset__Idl__subset,axiom,
! [I2: set_nat,H2: set_nat] :
( ( ord_less_eq_set_nat @ I2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ H2 @ I2 )
=> ( ord_less_eq_set_nat @ ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ H2 ) @ ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ I2 ) ) ) ) ).
% s.subset_Idl_subset
thf(fact_500_s_Ogenideal__self,axiom,
! [S: set_nat] :
( ( ord_less_eq_set_nat @ S @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ord_less_eq_set_nat @ S @ ( genide4496968333291595629t_unit @ ( mod_ring @ n ) @ S ) ) ) ).
% s.genideal_self
thf(fact_501_subset__empty,axiom,
! [A3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
= ( A3 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_502_empty__subsetI,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A3 ) ).
% empty_subsetI
thf(fact_503_insert__subset,axiom,
! [X: nat > set_int,A3: set_nat_set_int,B5: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ ( insert_nat_set_int @ X @ A3 ) @ B5 )
= ( ( member_nat_set_int @ X @ B5 )
& ( ord_le5995675665013768039et_int @ A3 @ B5 ) ) ) ).
% insert_subset
thf(fact_504_insert__subset,axiom,
! [X: set_nat,A3: set_set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A3 ) @ B5 )
= ( ( member_set_nat @ X @ B5 )
& ( ord_le6893508408891458716et_nat @ A3 @ B5 ) ) ) ).
% insert_subset
thf(fact_505_insert__subset,axiom,
! [X: nat,A3: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A3 ) @ B5 )
= ( ( member_nat @ X @ B5 )
& ( ord_less_eq_set_nat @ A3 @ B5 ) ) ) ).
% insert_subset
thf(fact_506_singletonI,axiom,
! [A: nat > set_int] : ( member_nat_set_int @ A @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ) ).
% singletonI
thf(fact_507_singletonI,axiom,
! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_508_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_509_units__of__units,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( units_1133518744711867743t_unit @ ( units_2222427131560811679t_unit @ G2 ) )
= ( units_1295200668666280182t_unit @ G2 ) ) ).
% units_of_units
thf(fact_510_units__of__mult,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( mult_n7532461433716462410t_unit @ ( units_2222427131560811679t_unit @ G2 ) )
= ( mult_n6028127365542633569t_unit @ G2 ) ) ).
% units_of_mult
thf(fact_511_units__of__one,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( one_nat_Product_unit @ ( units_2222427131560811679t_unit @ G2 ) )
= ( one_na902338870878123981t_unit @ G2 ) ) ).
% units_of_one
thf(fact_512_units__of__carrier,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( partia5989097197943085638t_unit @ ( units_2222427131560811679t_unit @ G2 ) )
= ( units_1295200668666280182t_unit @ G2 ) ) ).
% units_of_carrier
thf(fact_513_monoid_Ounits__group,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( group_1007331108230521615t_unit @ ( units_2222427131560811679t_unit @ G2 ) ) ) ).
% monoid.units_group
thf(fact_514_in__mono,axiom,
! [A3: set_nat_set_int,B5: set_nat_set_int,X: nat > set_int] :
( ( ord_le5995675665013768039et_int @ A3 @ B5 )
=> ( ( member_nat_set_int @ X @ A3 )
=> ( member_nat_set_int @ X @ B5 ) ) ) ).
% in_mono
thf(fact_515_in__mono,axiom,
! [A3: set_set_nat,B5: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B5 )
=> ( ( member_set_nat @ X @ A3 )
=> ( member_set_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_516_in__mono,axiom,
! [A3: set_nat,B5: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
=> ( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B5 ) ) ) ).
% in_mono
thf(fact_517_subsetD,axiom,
! [A3: set_nat_set_int,B5: set_nat_set_int,C: nat > set_int] :
( ( ord_le5995675665013768039et_int @ A3 @ B5 )
=> ( ( member_nat_set_int @ C @ A3 )
=> ( member_nat_set_int @ C @ B5 ) ) ) ).
% subsetD
thf(fact_518_subsetD,axiom,
! [A3: set_set_nat,B5: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B5 )
=> ( ( member_set_nat @ C @ A3 )
=> ( member_set_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_519_subsetD,axiom,
! [A3: set_nat,B5: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_520_equalityE,axiom,
! [A3: set_nat,B5: set_nat] :
( ( A3 = B5 )
=> ~ ( ( ord_less_eq_set_nat @ A3 @ B5 )
=> ~ ( ord_less_eq_set_nat @ B5 @ A3 ) ) ) ).
% equalityE
thf(fact_521_subset__eq,axiom,
( ord_le5995675665013768039et_int
= ( ^ [A6: set_nat_set_int,B6: set_nat_set_int] :
! [X3: nat > set_int] :
( ( member_nat_set_int @ X3 @ A6 )
=> ( member_nat_set_int @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_522_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A6: set_set_nat,B6: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A6 )
=> ( member_set_nat @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_523_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( member_nat @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_524_equalityD1,axiom,
! [A3: set_nat,B5: set_nat] :
( ( A3 = B5 )
=> ( ord_less_eq_set_nat @ A3 @ B5 ) ) ).
% equalityD1
thf(fact_525_equalityD2,axiom,
! [A3: set_nat,B5: set_nat] :
( ( A3 = B5 )
=> ( ord_less_eq_set_nat @ B5 @ A3 ) ) ).
% equalityD2
thf(fact_526_subset__iff,axiom,
( ord_le5995675665013768039et_int
= ( ^ [A6: set_nat_set_int,B6: set_nat_set_int] :
! [T: nat > set_int] :
( ( member_nat_set_int @ T @ A6 )
=> ( member_nat_set_int @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_527_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A6: set_set_nat,B6: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A6 )
=> ( member_set_nat @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_528_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A6 )
=> ( member_nat @ T @ B6 ) ) ) ) ).
% subset_iff
thf(fact_529_subset__refl,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).
% subset_refl
thf(fact_530_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_531_subset__trans,axiom,
! [A3: set_nat,B5: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ C3 )
=> ( ord_less_eq_set_nat @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_532_set__eq__subset,axiom,
( ( ^ [Y6: set_nat,Z4: set_nat] : ( Y6 = Z4 ) )
= ( ^ [A6: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_533_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_534_emptyE,axiom,
! [A: nat > set_int] :
~ ( member_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ).
% emptyE
thf(fact_535_emptyE,axiom,
! [A: set_nat] :
~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_536_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_537_equals0D,axiom,
! [A3: set_nat_set_int,A: nat > set_int] :
( ( A3 = bot_bo8417611410066262939et_int )
=> ~ ( member_nat_set_int @ A @ A3 ) ) ).
% equals0D
thf(fact_538_equals0D,axiom,
! [A3: set_set_nat,A: set_nat] :
( ( A3 = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_539_equals0D,axiom,
! [A3: set_nat,A: nat] :
( ( A3 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_540_equals0I,axiom,
! [A3: set_nat_set_int] :
( ! [Y3: nat > set_int] :
~ ( member_nat_set_int @ Y3 @ A3 )
=> ( A3 = bot_bo8417611410066262939et_int ) ) ).
% equals0I
thf(fact_541_equals0I,axiom,
! [A3: set_set_nat] :
( ! [Y3: set_nat] :
~ ( member_set_nat @ Y3 @ A3 )
=> ( A3 = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_542_equals0I,axiom,
! [A3: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A3 )
=> ( A3 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_543_ex__in__conv,axiom,
! [A3: set_nat_set_int] :
( ( ? [X3: nat > set_int] : ( member_nat_set_int @ X3 @ A3 ) )
= ( A3 != bot_bo8417611410066262939et_int ) ) ).
% ex_in_conv
thf(fact_544_ex__in__conv,axiom,
! [A3: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A3 ) )
= ( A3 != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_545_ex__in__conv,axiom,
! [A3: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A3 ) )
= ( A3 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_546_ring_Ogenideal__self,axiom,
! [R2: partia4934656038542163276t_unit,S: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ord_le4403425263959731960et_int @ S @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ord_le4403425263959731960et_int @ S @ ( genide1545711809618862555t_unit @ R2 @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_547_ring_Ogenideal__self,axiom,
! [R2: partia4692342223508353374t_unit,S: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ord_less_eq_set_nat @ S @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ord_less_eq_set_nat @ S @ ( genide4496968333291595629t_unit @ R2 @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_548_ring_Osubset__Idl__subset,axiom,
! [R2: partia4934656038542163276t_unit,I2: set_set_int,H2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( ord_le4403425263959731960et_int @ H2 @ I2 )
=> ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ R2 @ H2 ) @ ( genide1545711809618862555t_unit @ R2 @ I2 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_549_ring_Osubset__Idl__subset,axiom,
! [R2: partia4692342223508353374t_unit,I2: set_nat,H2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ord_less_eq_set_nat @ I2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( ord_less_eq_set_nat @ H2 @ I2 )
=> ( ord_less_eq_set_nat @ ( genide4496968333291595629t_unit @ R2 @ H2 ) @ ( genide4496968333291595629t_unit @ R2 @ I2 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_550_ring_Ogenideal__self_H,axiom,
! [R2: partia4934656038542163276t_unit,I: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ I @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ I @ ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ I @ bot_bot_set_set_int ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_551_ring_Ogenideal__self_H,axiom,
! [R2: partia4692342223508353374t_unit,I: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ I @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ I @ ( genide4496968333291595629t_unit @ R2 @ ( insert_nat @ I @ bot_bot_set_nat ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_552_ring_Ogenideal__zero,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) ) ).
% ring.genideal_zero
thf(fact_553_ring_Ogenideal__zero,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( genide4496968333291595629t_unit @ R2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) ) ).
% ring.genideal_zero
thf(fact_554_principalideal_Ogenerate,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( princi8860937869964495385t_unit @ I2 @ R2 )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
& ( I2
= ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ X4 @ bot_bot_set_set_int ) ) ) ) ) ).
% principalideal.generate
thf(fact_555_principalideal_Ogenerate,axiom,
! [I2: set_nat,R2: partia4692342223508353374t_unit] :
( ( princi4652470909602072491t_unit @ I2 @ R2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
& ( I2
= ( genide4496968333291595629t_unit @ R2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% principalideal.generate
thf(fact_556_ring_OIdl__subset__ideal_H,axiom,
! [R2: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) @ ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) )
= ( member_set_int @ A @ ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_557_ring_OIdl__subset__ideal_H,axiom,
! [R2: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( ord_less_eq_set_nat @ ( genide4496968333291595629t_unit @ R2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( genide4496968333291595629t_unit @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( member_nat @ A @ ( genide4496968333291595629t_unit @ R2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_558_ring_Ogenideal__one,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ ( one_se8065767436706823081t_unit @ R2 ) @ bot_bot_set_set_int ) )
= ( partia966996272515721803t_unit @ R2 ) ) ) ).
% ring.genideal_one
thf(fact_559_ring_Ogenideal__one,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( genide4496968333291595629t_unit @ R2 @ ( insert_nat @ ( one_na902338870878123981t_unit @ R2 ) @ bot_bot_set_nat ) )
= ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% ring.genideal_one
thf(fact_560_subset__insertI2,axiom,
! [A3: set_nat,B5: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
=> ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B @ B5 ) ) ) ).
% subset_insertI2
thf(fact_561_subset__insertI,axiom,
! [B5: set_nat,A: nat] : ( ord_less_eq_set_nat @ B5 @ ( insert_nat @ A @ B5 ) ) ).
% subset_insertI
thf(fact_562_subset__insert,axiom,
! [X: nat > set_int,A3: set_nat_set_int,B5: set_nat_set_int] :
( ~ ( member_nat_set_int @ X @ A3 )
=> ( ( ord_le5995675665013768039et_int @ A3 @ ( insert_nat_set_int @ X @ B5 ) )
= ( ord_le5995675665013768039et_int @ A3 @ B5 ) ) ) ).
% subset_insert
thf(fact_563_subset__insert,axiom,
! [X: set_nat,A3: set_set_nat,B5: set_set_nat] :
( ~ ( member_set_nat @ X @ A3 )
=> ( ( ord_le6893508408891458716et_nat @ A3 @ ( insert_set_nat @ X @ B5 ) )
= ( ord_le6893508408891458716et_nat @ A3 @ B5 ) ) ) ).
% subset_insert
thf(fact_564_subset__insert,axiom,
! [X: nat,A3: set_nat,B5: set_nat] :
( ~ ( member_nat @ X @ A3 )
=> ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X @ B5 ) )
= ( ord_less_eq_set_nat @ A3 @ B5 ) ) ) ).
% subset_insert
thf(fact_565_insert__mono,axiom,
! [C3: set_nat,D: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C3 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C3 ) @ ( insert_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_566_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_567_insert__not__empty,axiom,
! [A: nat,A3: set_nat] :
( ( insert_nat @ A @ A3 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_568_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_569_singleton__iff,axiom,
! [B: nat > set_int,A: nat > set_int] :
( ( member_nat_set_int @ B @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_570_singleton__iff,axiom,
! [B: set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_571_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_572_singletonD,axiom,
! [B: nat > set_int,A: nat > set_int] :
( ( member_nat_set_int @ B @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_573_singletonD,axiom,
! [B: set_nat,A: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_574_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_575_subset__singleton__iff,axiom,
! [X5: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_576_subset__singletonD,axiom,
! [A3: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A3 = bot_bot_set_nat )
| ( A3
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_577_s_Oset__add__zero,axiom,
! [A3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) @ A3 )
= A3 ) ) ).
% s.set_add_zero
thf(fact_578_abelian__group_Oa__lcos__m__assoc,axiom,
! [G2: partia4934656038542163276t_unit,M2: set_set_int,G: set_int,H: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ G @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ H @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( a_l_co3504123944629134560t_unit @ G2 @ G @ ( a_l_co3504123944629134560t_unit @ G2 @ H @ M2 ) )
= ( a_l_co3504123944629134560t_unit @ G2 @ ( add_se5859248395121729892t_unit @ G2 @ G @ H ) @ M2 ) ) ) ) ) ) ).
% abelian_group.a_lcos_m_assoc
thf(fact_579_abelian__group_Oa__lcos__m__assoc,axiom,
! [G2: partia4692342223508353374t_unit,M2: set_nat,G: nat,H: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ G @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ H @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( a_l_co3340896127261556338t_unit @ G2 @ G @ ( a_l_co3340896127261556338t_unit @ G2 @ H @ M2 ) )
= ( a_l_co3340896127261556338t_unit @ G2 @ ( add_nat_Product_unit @ G2 @ G @ H ) @ M2 ) ) ) ) ) ) ).
% abelian_group.a_lcos_m_assoc
thf(fact_580_abelian__group_Oa__lcos__mult__one,axiom,
! [G2: partia4934656038542163276t_unit,M2: set_set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( a_l_co3504123944629134560t_unit @ G2 @ ( zero_s6269048424454532197t_unit @ G2 ) @ M2 )
= M2 ) ) ) ).
% abelian_group.a_lcos_mult_one
thf(fact_581_abelian__group_Oa__lcos__mult__one,axiom,
! [G2: partia4692342223508353374t_unit,M2: set_nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( a_l_co3340896127261556338t_unit @ G2 @ ( zero_n5149899317435570679t_unit @ G2 ) @ M2 )
= M2 ) ) ) ).
% abelian_group.a_lcos_mult_one
thf(fact_582_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_583_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_584_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_585_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_586_s_Oadd_Oone__in__subset,axiom,
! [H2: set_nat] :
( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( H2 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ H2 )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X4 ) @ H2 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ H2 )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ H2 )
=> ( member_nat @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X4 @ Xa2 ) @ H2 ) ) )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ H2 ) ) ) ) ) ).
% s.add.one_in_subset
thf(fact_587_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_588_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_589_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_590_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_591_s_Oa__transpose__inv,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y )
= Z )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) @ Z )
= Y ) ) ) ) ) ).
% s.a_transpose_inv
thf(fact_592_s_Oadd_Oinv__mult__group,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ Y ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) ) ) ) ) ).
% s.add.inv_mult_group
thf(fact_593_s_Oadd_Oinv__solve__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( A
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ B ) @ C ) )
= ( C
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ B @ A ) ) ) ) ) ) ).
% s.add.inv_solve_left
thf(fact_594_s_Oadd_Oinv__solve__left_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ B ) @ C )
= A )
= ( C
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ B @ A ) ) ) ) ) ) ).
% s.add.inv_solve_left'
thf(fact_595_s_Oadd_Oinv__solve__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( A
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ B @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ C ) ) )
= ( B
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ A @ C ) ) ) ) ) ) ).
% s.add.inv_solve_right
thf(fact_596_s_Oadd_Oinv__solve__right_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ B @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ C ) )
= A )
= ( B
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ A @ C ) ) ) ) ) ) ).
% s.add.inv_solve_right'
thf(fact_597_s_Ominus__add,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ Y ) ) ) ) ) ).
% s.minus_add
thf(fact_598_s_Or__neg1,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ Y ) )
= Y ) ) ) ).
% s.r_neg1
thf(fact_599_s_Or__neg2,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) @ Y ) )
= Y ) ) ) ).
% s.r_neg2
thf(fact_600_s_Ol__minus,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) @ Y )
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y ) ) ) ) ) ).
% s.l_minus
thf(fact_601_s_Or__minus,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ Y ) )
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y ) ) ) ) ) ).
% s.r_minus
thf(fact_602_s_Oadd_Onat__pow__inv,axiom,
! [X: nat,I: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ I @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) )
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ I @ X ) ) ) ) ).
% s.add.nat_pow_inv
thf(fact_603_s_Oset__add__closed,axiom,
! [A3: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ B5 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ord_less_eq_set_nat @ ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ A3 @ B5 ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.set_add_closed
thf(fact_604_s_Oset__add__comm,axiom,
! [I2: set_nat,J2: set_nat] :
( ( ord_less_eq_set_nat @ I2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ J2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ I2 @ J2 )
= ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ J2 @ I2 ) ) ) ) ).
% s.set_add_comm
thf(fact_605_s_Osetadd__subset__G,axiom,
! [H2: set_nat,K2: set_nat] :
( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ord_less_eq_set_nat @ ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ H2 @ K2 ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.setadd_subset_G
thf(fact_606_s_Oadd_Oint__pow__inv,axiom,
! [X: nat,I: int] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) )
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ X ) ) ) ) ).
% s.add.int_pow_inv
thf(fact_607_s_Ol__neg,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.l_neg
thf(fact_608_s_Ominus__equality,axiom,
! [Y: nat,X: nat] :
( ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ Y @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X )
= Y ) ) ) ) ).
% s.minus_equality
thf(fact_609_s_Or__neg,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.r_neg
thf(fact_610_s_Oadd_Oinv__closed,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.add.inv_closed
thf(fact_611_s_Ominus__minus,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X ) )
= X ) ) ).
% s.minus_minus
thf(fact_612_s_Ominus__zero,axiom,
( ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ).
% s.minus_zero
thf(fact_613_s_Oadd_Oinv__eq__1__iff,axiom,
! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
= ( X
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.add.inv_eq_1_iff
thf(fact_614_s_OUnits__minus__one__closed,axiom,
member_nat @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) ).
% s.Units_minus_one_closed
thf(fact_615_ring_Oring__simprules_I3_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R2 @ X ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_616_ring_Oring__simprules_I3_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ R2 @ X ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_617_ring_Oring__simprules_I20_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( a_inv_5951419416477254493t_unit @ R2 @ ( a_inv_5951419416477254493t_unit @ R2 @ X ) )
= X ) ) ) ).
% ring.ring_simprules(20)
thf(fact_618_ring_Oring__simprules_I20_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( a_inv_2472168910397739247t_unit @ R2 @ ( a_inv_2472168910397739247t_unit @ R2 @ X ) )
= X ) ) ) ).
% ring.ring_simprules(20)
thf(fact_619_ring_Ominus__zero,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( a_inv_5951419416477254493t_unit @ R2 @ ( zero_s6269048424454532197t_unit @ R2 ) )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ).
% ring.minus_zero
thf(fact_620_ring_Ominus__zero,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( a_inv_2472168910397739247t_unit @ R2 @ ( zero_n5149899317435570679t_unit @ R2 ) )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ).
% ring.minus_zero
thf(fact_621_abelian__group_Oa__transpose__inv,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int,Z: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( ( add_se5859248395121729892t_unit @ G2 @ X @ Y )
= Z )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( add_se5859248395121729892t_unit @ G2 @ ( a_inv_5951419416477254493t_unit @ G2 @ X ) @ Z )
= Y ) ) ) ) ) ) ).
% abelian_group.a_transpose_inv
thf(fact_622_abelian__group_Oa__transpose__inv,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat,Z: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( ( add_nat_Product_unit @ G2 @ X @ Y )
= Z )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( add_nat_Product_unit @ G2 @ ( a_inv_2472168910397739247t_unit @ G2 @ X ) @ Z )
= Y ) ) ) ) ) ) ).
% abelian_group.a_transpose_inv
thf(fact_623_abelian__group_Oa__inv__closed,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ G2 @ X ) @ ( partia966996272515721803t_unit @ G2 ) ) ) ) ).
% abelian_group.a_inv_closed
thf(fact_624_abelian__group_Oa__inv__closed,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ G2 @ X ) @ ( partia3499330772048238685t_unit @ G2 ) ) ) ) ).
% abelian_group.a_inv_closed
thf(fact_625_abelian__group_Ominus__minus,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( a_inv_5951419416477254493t_unit @ G2 @ ( a_inv_5951419416477254493t_unit @ G2 @ X ) )
= X ) ) ) ).
% abelian_group.minus_minus
thf(fact_626_abelian__group_Ominus__minus,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( a_inv_2472168910397739247t_unit @ G2 @ ( a_inv_2472168910397739247t_unit @ G2 @ X ) )
= X ) ) ) ).
% abelian_group.minus_minus
thf(fact_627_abelian__group_Osetadd__subset__G,axiom,
! [G2: partia4934656038542163276t_unit,H2: set_set_int,K2: set_set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( ord_le4403425263959731960et_int @ K2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ G2 @ H2 @ K2 ) @ ( partia966996272515721803t_unit @ G2 ) ) ) ) ) ).
% abelian_group.setadd_subset_G
thf(fact_628_abelian__group_Osetadd__subset__G,axiom,
! [G2: partia4692342223508353374t_unit,H2: set_nat,K2: set_nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ord_less_eq_set_nat @ ( set_ad320919470248169786t_unit @ G2 @ H2 @ K2 ) @ ( partia3499330772048238685t_unit @ G2 ) ) ) ) ) ).
% abelian_group.setadd_subset_G
thf(fact_629_abelian__monoid_Oset__add__closed,axiom,
! [G2: partia4934656038542163276t_unit,A3: set_set_int,B5: set_set_int] :
( ( abelia3815030880812984441t_unit @ G2 )
=> ( ( ord_le4403425263959731960et_int @ A3 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( ord_le4403425263959731960et_int @ B5 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ G2 @ A3 @ B5 ) @ ( partia966996272515721803t_unit @ G2 ) ) ) ) ) ).
% abelian_monoid.set_add_closed
thf(fact_630_abelian__monoid_Oset__add__closed,axiom,
! [G2: partia4692342223508353374t_unit,A3: set_nat,B5: set_nat] :
( ( abelia362511065248671243t_unit @ G2 )
=> ( ( ord_less_eq_set_nat @ A3 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( ord_less_eq_set_nat @ B5 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ord_less_eq_set_nat @ ( set_ad320919470248169786t_unit @ G2 @ A3 @ B5 ) @ ( partia3499330772048238685t_unit @ G2 ) ) ) ) ) ).
% abelian_monoid.set_add_closed
thf(fact_631_ring_Oring__simprules_I17_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ ( add_se5859248395121729892t_unit @ R2 @ ( a_inv_5951419416477254493t_unit @ R2 @ X ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_632_ring_Oring__simprules_I17_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ ( add_nat_Product_unit @ R2 @ ( a_inv_2472168910397739247t_unit @ R2 @ X ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_633_ring_Oring__simprules_I18_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( a_inv_5951419416477254493t_unit @ R2 @ X ) @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_634_ring_Oring__simprules_I18_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( a_inv_2472168910397739247t_unit @ R2 @ X ) @ ( add_nat_Product_unit @ R2 @ X @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_635_ring_Oring__simprules_I19_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( a_inv_5951419416477254493t_unit @ R2 @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ R2 @ ( a_inv_5951419416477254493t_unit @ R2 @ X ) @ ( a_inv_5951419416477254493t_unit @ R2 @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_636_ring_Oring__simprules_I19_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( a_inv_2472168910397739247t_unit @ R2 @ ( add_nat_Product_unit @ R2 @ X @ Y ) )
= ( add_nat_Product_unit @ R2 @ ( a_inv_2472168910397739247t_unit @ R2 @ X ) @ ( a_inv_2472168910397739247t_unit @ R2 @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_637_ring_Or__minus,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ X @ ( a_inv_5951419416477254493t_unit @ R2 @ Y ) )
= ( a_inv_5951419416477254493t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Y ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_638_ring_Or__minus,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ X @ ( a_inv_2472168910397739247t_unit @ R2 @ Y ) )
= ( a_inv_2472168910397739247t_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_639_ring_Ol__minus,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( a_inv_5951419416477254493t_unit @ R2 @ X ) @ Y )
= ( a_inv_5951419416477254493t_unit @ R2 @ ( mult_s3864001451298473021t_unit @ R2 @ X @ Y ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_640_ring_Ol__minus,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( a_inv_2472168910397739247t_unit @ R2 @ X ) @ Y )
= ( a_inv_2472168910397739247t_unit @ R2 @ ( mult_n6028127365542633569t_unit @ R2 @ X @ Y ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_641_abelian__group_Ominus__add,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( a_inv_5951419416477254493t_unit @ G2 @ ( add_se5859248395121729892t_unit @ G2 @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ G2 @ ( a_inv_5951419416477254493t_unit @ G2 @ X ) @ ( a_inv_5951419416477254493t_unit @ G2 @ Y ) ) ) ) ) ) ).
% abelian_group.minus_add
thf(fact_642_abelian__group_Ominus__add,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( a_inv_2472168910397739247t_unit @ G2 @ ( add_nat_Product_unit @ G2 @ X @ Y ) )
= ( add_nat_Product_unit @ G2 @ ( a_inv_2472168910397739247t_unit @ G2 @ X ) @ ( a_inv_2472168910397739247t_unit @ G2 @ Y ) ) ) ) ) ) ).
% abelian_group.minus_add
thf(fact_643_abelian__group_Or__neg2,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( add_se5859248395121729892t_unit @ G2 @ X @ ( add_se5859248395121729892t_unit @ G2 @ ( a_inv_5951419416477254493t_unit @ G2 @ X ) @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg2
thf(fact_644_abelian__group_Or__neg2,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( add_nat_Product_unit @ G2 @ X @ ( add_nat_Product_unit @ G2 @ ( a_inv_2472168910397739247t_unit @ G2 @ X ) @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg2
thf(fact_645_abelian__group_Or__neg1,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( add_se5859248395121729892t_unit @ G2 @ ( a_inv_5951419416477254493t_unit @ G2 @ X ) @ ( add_se5859248395121729892t_unit @ G2 @ X @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg1
thf(fact_646_abelian__group_Or__neg1,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( add_nat_Product_unit @ G2 @ ( a_inv_2472168910397739247t_unit @ G2 @ X ) @ ( add_nat_Product_unit @ G2 @ X @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg1
thf(fact_647_ring_OUnits__minus__one__closed,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R2 @ ( one_se8065767436706823081t_unit @ R2 ) ) @ ( units_4038138251425117394t_unit @ R2 ) ) ) ).
% ring.Units_minus_one_closed
thf(fact_648_ring_OUnits__minus__one__closed,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ R2 @ ( one_na902338870878123981t_unit @ R2 ) ) @ ( units_1295200668666280182t_unit @ R2 ) ) ) ).
% ring.Units_minus_one_closed
thf(fact_649_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_650_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_651_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_652_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_653_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_654_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_655_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B7: int] : ( plus_plus_int @ B7 @ A4 ) ) ) ).
% add.commute
thf(fact_656_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B7: nat] : ( plus_plus_nat @ B7 @ A4 ) ) ) ).
% add.commute
thf(fact_657_add_Oright__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_658_add_Oleft__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_659_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_660_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_661_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_662_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_663_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_664_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_665_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_666_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_667_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_668_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_669_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_670_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_671_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B7: int] : ( times_times_int @ B7 @ A4 ) ) ) ).
% mult.commute
thf(fact_672_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B7: nat] : ( times_times_nat @ B7 @ A4 ) ) ) ).
% mult.commute
thf(fact_673_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_674_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_675_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_676_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_677_ring_Oset__add__comm,axiom,
! [R2: partia4934656038542163276t_unit,I2: set_set_int,J2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( ord_le4403425263959731960et_int @ J2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( set_ad273131178244904872t_unit @ R2 @ I2 @ J2 )
= ( set_ad273131178244904872t_unit @ R2 @ J2 @ I2 ) ) ) ) ) ).
% ring.set_add_comm
thf(fact_678_ring_Oset__add__comm,axiom,
! [R2: partia4692342223508353374t_unit,I2: set_nat,J2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ord_less_eq_set_nat @ I2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( ord_less_eq_set_nat @ J2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( set_ad320919470248169786t_unit @ R2 @ I2 @ J2 )
= ( set_ad320919470248169786t_unit @ R2 @ J2 @ I2 ) ) ) ) ) ).
% ring.set_add_comm
thf(fact_679_ring_Oring__simprules_I9_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ ( a_inv_5951419416477254493t_unit @ R2 @ X ) @ X )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_680_ring_Oring__simprules_I9_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ ( a_inv_2472168910397739247t_unit @ R2 @ X ) @ X )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_681_ring_Oring__simprules_I16_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( add_se5859248395121729892t_unit @ R2 @ X @ ( a_inv_5951419416477254493t_unit @ R2 @ X ) )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_682_ring_Oring__simprules_I16_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( add_nat_Product_unit @ R2 @ X @ ( a_inv_2472168910397739247t_unit @ R2 @ X ) )
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_683_abelian__group_Ol__neg,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( add_se5859248395121729892t_unit @ G2 @ ( a_inv_5951419416477254493t_unit @ G2 @ X ) @ X )
= ( zero_s6269048424454532197t_unit @ G2 ) ) ) ) ).
% abelian_group.l_neg
thf(fact_684_abelian__group_Ol__neg,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( add_nat_Product_unit @ G2 @ ( a_inv_2472168910397739247t_unit @ G2 @ X ) @ X )
= ( zero_n5149899317435570679t_unit @ G2 ) ) ) ) ).
% abelian_group.l_neg
thf(fact_685_abelian__group_Or__neg,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( add_se5859248395121729892t_unit @ G2 @ X @ ( a_inv_5951419416477254493t_unit @ G2 @ X ) )
= ( zero_s6269048424454532197t_unit @ G2 ) ) ) ) ).
% abelian_group.r_neg
thf(fact_686_abelian__group_Or__neg,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( add_nat_Product_unit @ G2 @ X @ ( a_inv_2472168910397739247t_unit @ G2 @ X ) )
= ( zero_n5149899317435570679t_unit @ G2 ) ) ) ) ).
% abelian_group.r_neg
thf(fact_687_abelian__group_Ominus__equality,axiom,
! [G2: partia4934656038542163276t_unit,Y: set_int,X: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( ( add_se5859248395121729892t_unit @ G2 @ Y @ X )
= ( zero_s6269048424454532197t_unit @ G2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( a_inv_5951419416477254493t_unit @ G2 @ X )
= Y ) ) ) ) ) ).
% abelian_group.minus_equality
thf(fact_688_abelian__group_Ominus__equality,axiom,
! [G2: partia4692342223508353374t_unit,Y: nat,X: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( ( add_nat_Product_unit @ G2 @ Y @ X )
= ( zero_n5149899317435570679t_unit @ G2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( a_inv_2472168910397739247t_unit @ G2 @ X )
= Y ) ) ) ) ) ).
% abelian_group.minus_equality
thf(fact_689_add__le__imp__le__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_imp_le_right
thf(fact_690_add__le__imp__le__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_imp_le_right
thf(fact_691_add__le__imp__le__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_imp_le_left
thf(fact_692_add__le__imp__le__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_imp_le_left
thf(fact_693_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B7: nat] :
? [C4: nat] :
( B7
= ( plus_plus_nat @ A4 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_694_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_695_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_696_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_697_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_698_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_699_add__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 @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_700_add__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 @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_701_add__mono__thms__linordered__semiring_I1_J,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_mono_thms_linordered_semiring(1)
thf(fact_702_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_703_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_704_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_705_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_706_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_707_ring_Oset__add__zero,axiom,
! [R2: partia4934656038542163276t_unit,A3: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ord_le4403425263959731960et_int @ A3 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( set_ad273131178244904872t_unit @ R2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) @ A3 )
= A3 ) ) ) ).
% ring.set_add_zero
thf(fact_708_ring_Oset__add__zero,axiom,
! [R2: partia4692342223508353374t_unit,A3: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ord_less_eq_set_nat @ A3 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( set_ad320919470248169786t_unit @ R2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) @ A3 )
= A3 ) ) ) ).
% ring.set_add_zero
thf(fact_709_abelian__group_Oa__l__coset__subset__G,axiom,
! [G2: partia4934656038542163276t_unit,H2: set_set_int,X: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ord_le4403425263959731960et_int @ ( a_l_co3504123944629134560t_unit @ G2 @ X @ H2 ) @ ( partia966996272515721803t_unit @ G2 ) ) ) ) ) ).
% abelian_group.a_l_coset_subset_G
thf(fact_710_abelian__group_Oa__l__coset__subset__G,axiom,
! [G2: partia4692342223508353374t_unit,H2: set_nat,X: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ord_less_eq_set_nat @ ( a_l_co3340896127261556338t_unit @ G2 @ X @ H2 ) @ ( partia3499330772048238685t_unit @ G2 ) ) ) ) ) ).
% abelian_group.a_l_coset_subset_G
thf(fact_711_s_Ominus__eq,axiom,
! [X: nat,Y: nat] :
( ( a_minu1751788497103602224t_unit @ ( mod_ring @ n ) @ X @ Y )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ Y ) ) ) ).
% s.minus_eq
thf(fact_712_s_Oadd_Oint__pow__neg__int,axiom,
! [X: nat,N: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ X )
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) ) ) ) ).
% s.add.int_pow_neg_int
thf(fact_713_s_OsubringI,axiom,
! [H2: set_nat] :
( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) @ H2 )
=> ( ! [H3: nat] :
( ( member_nat @ H3 @ H2 )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ H3 ) @ H2 ) )
=> ( ! [H1: nat,H22: nat] :
( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ H1 @ H22 ) @ H2 ) ) )
=> ( ! [H1: nat,H22: nat] :
( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( member_nat @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ H1 @ H22 ) @ H2 ) ) )
=> ( subrin2893992908230074586t_unit @ H2 @ ( mod_ring @ n ) ) ) ) ) ) ) ).
% s.subringI
thf(fact_714_s_Oadd__additive__subgroups,axiom,
! [H2: set_nat,K2: set_nat] :
( ( additi4297497278381856430t_unit @ H2 @ ( mod_ring @ n ) )
=> ( ( additi4297497278381856430t_unit @ K2 @ ( mod_ring @ n ) )
=> ( additi4297497278381856430t_unit @ ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ H2 @ K2 ) @ ( mod_ring @ n ) ) ) ) ).
% s.add_additive_subgroups
thf(fact_715_s_Oadd_Oint__pow__diff,axiom,
! [X: nat,N: int,M: int] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ ( minus_minus_int @ N @ M ) @ X )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ N @ X ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ M @ X ) ) ) ) ) ).
% s.add.int_pow_diff
thf(fact_716_s_Ocarrier__is__subring,axiom,
subrin2893992908230074586t_unit @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) @ ( mod_ring @ n ) ).
% s.carrier_is_subring
thf(fact_717_s_Oadd_Oint__pow__neg,axiom,
! [X: nat,I: int] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ ( uminus_uminus_int @ I ) @ X )
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( add_po2420080144553951285it_int @ ( mod_ring @ n ) @ I @ X ) ) ) ) ).
% s.add.int_pow_neg
thf(fact_718_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_719_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_720_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_721_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_722_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_723_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_724_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_725_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_726_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_727_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_728_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_729_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_730_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_731_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_732_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_733_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_734_s_Ominus__closed,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( a_minu1751788497103602224t_unit @ ( mod_ring @ n ) @ X @ Y ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.minus_closed
thf(fact_735_s_Or__right__minus__eq,axiom,
! [A: nat,B: nat] :
( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( a_minu1751788497103602224t_unit @ ( mod_ring @ n ) @ A @ B )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
= ( A = B ) ) ) ) ).
% s.r_right_minus_eq
thf(fact_736_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B7: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B7 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_737_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B7: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B7 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_738_group__cancel_Osub2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B5 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_739_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_740_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_741_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_742_diff__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D2 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_743_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_744_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_745_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_746_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_747_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_748_group__cancel_Osub1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_749_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_750_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_751_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_752_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_753_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_754_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_755_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_756_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_757_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_758_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_759_group__cancel_Oneg1,axiom,
! [A3: int,K: int,A: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A3 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_760_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_761_additive__subgroup_Oa__Hcarr,axiom,
! [H2: set_set_int,G2: partia4934656038542163276t_unit,H: set_int] :
( ( additi7073586575563672860t_unit @ H2 @ G2 )
=> ( ( member_set_int @ H @ H2 )
=> ( member_set_int @ H @ ( partia966996272515721803t_unit @ G2 ) ) ) ) ).
% additive_subgroup.a_Hcarr
thf(fact_762_additive__subgroup_Oa__Hcarr,axiom,
! [H2: set_nat,G2: partia4692342223508353374t_unit,H: nat] :
( ( additi4297497278381856430t_unit @ H2 @ G2 )
=> ( ( member_nat @ H @ H2 )
=> ( member_nat @ H @ ( partia3499330772048238685t_unit @ G2 ) ) ) ) ).
% additive_subgroup.a_Hcarr
thf(fact_763_additive__subgroup_Ozero__closed,axiom,
! [H2: set_nat,G2: partia4692342223508353374t_unit] :
( ( additi4297497278381856430t_unit @ H2 @ G2 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ G2 ) @ H2 ) ) ).
% additive_subgroup.zero_closed
thf(fact_764_additive__subgroup_Oa__closed,axiom,
! [H2: set_nat,G2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( additi4297497278381856430t_unit @ H2 @ G2 )
=> ( ( member_nat @ X @ H2 )
=> ( ( member_nat @ Y @ H2 )
=> ( member_nat @ ( add_nat_Product_unit @ G2 @ X @ Y ) @ H2 ) ) ) ) ).
% additive_subgroup.a_closed
thf(fact_765_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_766_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_767_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_768_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_769_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_770_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_771_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_772_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_773_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_774_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_775_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_776_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_777_additive__subgroup_Oa__subset,axiom,
! [H2: set_set_int,G2: partia4934656038542163276t_unit] :
( ( additi7073586575563672860t_unit @ H2 @ G2 )
=> ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ G2 ) ) ) ).
% additive_subgroup.a_subset
thf(fact_778_additive__subgroup_Oa__subset,axiom,
! [H2: set_nat,G2: partia4692342223508353374t_unit] :
( ( additi4297497278381856430t_unit @ H2 @ G2 )
=> ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ G2 ) ) ) ).
% additive_subgroup.a_subset
thf(fact_779_ring_Oring__simprules_I4_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
=> ( member_set_int @ ( a_minu5974516859897376926t_unit @ R2 @ X @ Y ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_780_ring_Oring__simprules_I4_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( member_nat @ ( a_minu1751788497103602224t_unit @ R2 @ X @ Y ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_781_abelian__group_Oadd__additive__subgroups,axiom,
! [G2: partia4692342223508353374t_unit,H2: set_nat,K2: set_nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( additi4297497278381856430t_unit @ H2 @ G2 )
=> ( ( additi4297497278381856430t_unit @ K2 @ G2 )
=> ( additi4297497278381856430t_unit @ ( set_ad320919470248169786t_unit @ G2 @ H2 @ K2 ) @ G2 ) ) ) ) ).
% abelian_group.add_additive_subgroups
thf(fact_782_a__minus__def,axiom,
( a_minu1751788497103602224t_unit
= ( ^ [R3: partia4692342223508353374t_unit,X3: nat,Y4: nat] : ( add_nat_Product_unit @ R3 @ X3 @ ( a_inv_2472168910397739247t_unit @ R3 @ Y4 ) ) ) ) ).
% a_minus_def
thf(fact_783_abelian__group_Ominus__closed,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( member_set_int @ ( a_minu5974516859897376926t_unit @ G2 @ X @ Y ) @ ( partia966996272515721803t_unit @ G2 ) ) ) ) ) ).
% abelian_group.minus_closed
thf(fact_784_abelian__group_Ominus__closed,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( member_nat @ ( a_minu1751788497103602224t_unit @ G2 @ X @ Y ) @ ( partia3499330772048238685t_unit @ G2 ) ) ) ) ) ).
% abelian_group.minus_closed
thf(fact_785_ring_Oring__simprules_I14_J,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( a_minu5974516859897376926t_unit @ R2 @ X @ Y )
= ( add_se5859248395121729892t_unit @ R2 @ X @ ( a_inv_5951419416477254493t_unit @ R2 @ Y ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_786_ring_Oring__simprules_I14_J,axiom,
! [R2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( a_minu1751788497103602224t_unit @ R2 @ X @ Y )
= ( add_nat_Product_unit @ R2 @ X @ ( a_inv_2472168910397739247t_unit @ R2 @ Y ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_787_abelian__group_Ominus__eq,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( a_minu1751788497103602224t_unit @ G2 @ X @ Y )
= ( add_nat_Product_unit @ G2 @ X @ ( a_inv_2472168910397739247t_unit @ G2 @ Y ) ) ) ) ).
% abelian_group.minus_eq
thf(fact_788_ring_OsubringI,axiom,
! [R2: partia4934656038542163276t_unit,H2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ ( one_se8065767436706823081t_unit @ R2 ) @ H2 )
=> ( ! [H3: set_int] :
( ( member_set_int @ H3 @ H2 )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R2 @ H3 ) @ H2 ) )
=> ( ! [H1: set_int,H22: set_int] :
( ( member_set_int @ H1 @ H2 )
=> ( ( member_set_int @ H22 @ H2 )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ H1 @ H22 ) @ H2 ) ) )
=> ( ! [H1: set_int,H22: set_int] :
( ( member_set_int @ H1 @ H2 )
=> ( ( member_set_int @ H22 @ H2 )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R2 @ H1 @ H22 ) @ H2 ) ) )
=> ( subrin7689096310803670856t_unit @ H2 @ R2 ) ) ) ) ) ) ) ).
% ring.subringI
thf(fact_789_ring_OsubringI,axiom,
! [R2: partia4692342223508353374t_unit,H2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ ( one_na902338870878123981t_unit @ R2 ) @ H2 )
=> ( ! [H3: nat] :
( ( member_nat @ H3 @ H2 )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ R2 @ H3 ) @ H2 ) )
=> ( ! [H1: nat,H22: nat] :
( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ H1 @ H22 ) @ H2 ) ) )
=> ( ! [H1: nat,H22: nat] :
( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ H1 @ H22 ) @ H2 ) ) )
=> ( subrin2893992908230074586t_unit @ H2 @ R2 ) ) ) ) ) ) ) ).
% ring.subringI
thf(fact_790_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_791_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_792_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_793_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_794_s_OsubcringI,axiom,
! [H2: set_nat] :
( ( subrin2893992908230074586t_unit @ H2 @ ( mod_ring @ n ) )
=> ( ! [H1: nat,H22: nat] :
( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ H1 @ H22 )
= ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ H22 @ H1 ) ) ) )
=> ( subcri1627753237249443161t_unit @ H2 @ ( mod_ring @ n ) ) ) ) ).
% s.subcringI
thf(fact_795_Compl__subset__Compl__iff,axiom,
! [A3: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( uminus5710092332889474511et_nat @ B5 ) )
= ( ord_less_eq_set_nat @ B5 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_796_Compl__anti__mono,axiom,
! [A3: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B5 ) @ ( uminus5710092332889474511et_nat @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_797_Diff__empty,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ A3 @ bot_bot_set_nat )
= A3 ) ).
% Diff_empty
thf(fact_798_empty__Diff,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A3 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_799_Diff__cancel,axiom,
! [A3: set_nat] :
( ( minus_minus_set_nat @ A3 @ A3 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_800_s_Oadd_Opow__eq__div2,axiom,
! [X: nat,M: nat,N: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ M @ X )
= ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ N @ X ) )
=> ( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ ( minus_minus_nat @ M @ N ) @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.add.pow_eq_div2
thf(fact_801_Diff__eq__empty__iff,axiom,
! [A3: set_nat,B5: set_nat] :
( ( ( minus_minus_set_nat @ A3 @ B5 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A3 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_802_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_803_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_804_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_805_insert__Diff__single,axiom,
! [A: nat,A3: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_806_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_807_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_808_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_809_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_810_subset__Compl__singleton,axiom,
! [A3: set_nat_set_int,B: nat > set_int] :
( ( ord_le5995675665013768039et_int @ A3 @ ( uminus4718767861138198480et_int @ ( insert_nat_set_int @ B @ bot_bo8417611410066262939et_int ) ) )
= ( ~ ( member_nat_set_int @ B @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_811_subset__Compl__singleton,axiom,
! [A3: set_set_nat,B: set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
= ( ~ ( member_set_nat @ B @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_812_subset__Compl__singleton,axiom,
! [A3: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_813_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_814_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_815_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_816_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_817_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_818_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_819_Compl__insert,axiom,
! [X: nat,A3: set_nat] :
( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A3 ) )
= ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% Compl_insert
thf(fact_820_Diff__mono,axiom,
! [A3: set_nat,C3: set_nat,D: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ C3 )
=> ( ( ord_less_eq_set_nat @ D @ B5 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B5 ) @ ( minus_minus_set_nat @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_821_Diff__subset,axiom,
! [A3: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B5 ) @ A3 ) ).
% Diff_subset
thf(fact_822_double__diff,axiom,
! [A3: set_nat,B5: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
=> ( ( ord_less_eq_set_nat @ B5 @ C3 )
=> ( ( minus_minus_set_nat @ B5 @ ( minus_minus_set_nat @ C3 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_823_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_824_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_825_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_826_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_827_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_828_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_829_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_830_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_831_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_832_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_833_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_834_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_835_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_836_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_837_subcringE_I7_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H12: nat,H23: nat] :
( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subcringE(7)
thf(fact_838_subcringE_I4_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( H2 != bot_bot_set_nat ) ) ).
% subcringE(4)
thf(fact_839_subcringE_I2_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ H2 ) ) ).
% subcringE(2)
thf(fact_840_subcring_Osub__m__comm,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H12: nat,H23: nat] :
( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ H12 @ H23 )
= ( mult_n6028127365542633569t_unit @ R2 @ H23 @ H12 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_841_subcringE_I6_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H12: nat,H23: nat] :
( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subcringE(6)
thf(fact_842_subcringE_I3_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ R2 ) @ H2 ) ) ).
% subcringE(3)
thf(fact_843_subcringE_I1_J,axiom,
! [H2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( subcri1024317279029940167t_unit @ H2 @ R2 )
=> ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% subcringE(1)
thf(fact_844_subcringE_I1_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% subcringE(1)
thf(fact_845_subset__Compl__self__eq,axiom,
! [A3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_846_subset__Diff__insert,axiom,
! [A3: set_nat_set_int,B5: set_nat_set_int,X: nat > set_int,C3: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ A3 @ ( minus_3247115583872269408et_int @ B5 @ ( insert_nat_set_int @ X @ C3 ) ) )
= ( ( ord_le5995675665013768039et_int @ A3 @ ( minus_3247115583872269408et_int @ B5 @ C3 ) )
& ~ ( member_nat_set_int @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_847_subset__Diff__insert,axiom,
! [A3: set_set_nat,B5: set_set_nat,X: set_nat,C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ ( minus_2163939370556025621et_nat @ B5 @ ( insert_set_nat @ X @ C3 ) ) )
= ( ( ord_le6893508408891458716et_nat @ A3 @ ( minus_2163939370556025621et_nat @ B5 @ C3 ) )
& ~ ( member_set_nat @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_848_subset__Diff__insert,axiom,
! [A3: set_nat,B5: set_nat,X: nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B5 @ ( insert_nat @ X @ C3 ) ) )
= ( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B5 @ C3 ) )
& ~ ( member_nat @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_849_Diff__insert__absorb,axiom,
! [X: nat > set_int,A3: set_nat_set_int] :
( ~ ( member_nat_set_int @ X @ A3 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X @ A3 ) @ ( insert_nat_set_int @ X @ bot_bo8417611410066262939et_int ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_850_Diff__insert__absorb,axiom,
! [X: set_nat,A3: set_set_nat] :
( ~ ( member_set_nat @ X @ A3 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A3 ) @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_851_Diff__insert__absorb,axiom,
! [X: nat,A3: set_nat] :
( ~ ( member_nat @ X @ A3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A3 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_852_Diff__insert2,axiom,
! [A3: set_nat,A: nat,B5: set_nat] :
( ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ B5 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B5 ) ) ).
% Diff_insert2
thf(fact_853_insert__Diff,axiom,
! [A: nat > set_int,A3: set_nat_set_int] :
( ( member_nat_set_int @ A @ A3 )
=> ( ( insert_nat_set_int @ A @ ( minus_3247115583872269408et_int @ A3 @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_854_insert__Diff,axiom,
! [A: set_nat,A3: set_set_nat] :
( ( member_set_nat @ A @ A3 )
=> ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_855_insert__Diff,axiom,
! [A: nat,A3: set_nat] :
( ( member_nat @ A @ A3 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_856_Diff__insert,axiom,
! [A3: set_nat,A: nat,B5: set_nat] :
( ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ B5 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B5 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_857_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_858_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_859_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_860_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_861_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_862_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_863_ring_OsubcringI,axiom,
! [R2: partia4934656038542163276t_unit,H2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subrin7689096310803670856t_unit @ H2 @ R2 )
=> ( ! [H1: set_int,H22: set_int] :
( ( member_set_int @ H1 @ H2 )
=> ( ( member_set_int @ H22 @ H2 )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ H1 @ H22 )
= ( mult_s3864001451298473021t_unit @ R2 @ H22 @ H1 ) ) ) )
=> ( subcri1024317279029940167t_unit @ H2 @ R2 ) ) ) ) ).
% ring.subcringI
thf(fact_864_ring_OsubcringI,axiom,
! [R2: partia4692342223508353374t_unit,H2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subrin2893992908230074586t_unit @ H2 @ R2 )
=> ( ! [H1: nat,H22: nat] :
( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ H1 @ H22 )
= ( mult_n6028127365542633569t_unit @ R2 @ H22 @ H1 ) ) ) )
=> ( subcri1627753237249443161t_unit @ H2 @ R2 ) ) ) ) ).
% ring.subcringI
thf(fact_865_Diff__single__insert,axiom,
! [A3: set_nat,X: nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B5 )
=> ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_866_subset__insert__iff,axiom,
! [A3: set_nat_set_int,X: nat > set_int,B5: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ A3 @ ( insert_nat_set_int @ X @ B5 ) )
= ( ( ( member_nat_set_int @ X @ A3 )
=> ( ord_le5995675665013768039et_int @ ( minus_3247115583872269408et_int @ A3 @ ( insert_nat_set_int @ X @ bot_bo8417611410066262939et_int ) ) @ B5 ) )
& ( ~ ( member_nat_set_int @ X @ A3 )
=> ( ord_le5995675665013768039et_int @ A3 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_867_subset__insert__iff,axiom,
! [A3: set_set_nat,X: set_nat,B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ ( insert_set_nat @ X @ B5 ) )
= ( ( ( member_set_nat @ X @ A3 )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B5 ) )
& ( ~ ( member_set_nat @ X @ A3 )
=> ( ord_le6893508408891458716et_nat @ A3 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_868_subset__insert__iff,axiom,
! [A3: set_nat,X: nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X @ B5 ) )
= ( ( ( member_nat @ X @ A3 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B5 ) )
& ( ~ ( member_nat @ X @ A3 )
=> ( ord_less_eq_set_nat @ A3 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_869_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_870_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_871_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_872_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_873_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_874_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_875_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_876_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_877_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_878_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_879_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_880_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_881_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_882_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_883_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_884_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_885_minus__mult__commute,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% minus_mult_commute
thf(fact_886_square__eq__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ A )
= ( times_times_int @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ).
% square_eq_iff
thf(fact_887_subringE_I2_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subrin2893992908230074586t_unit @ H2 @ R2 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ H2 ) ) ).
% subringE(2)
thf(fact_888_subringE_I7_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H12: nat,H23: nat] :
( ( subrin2893992908230074586t_unit @ H2 @ R2 )
=> ( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subringE(7)
thf(fact_889_subringE_I4_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subrin2893992908230074586t_unit @ H2 @ R2 )
=> ( H2 != bot_bot_set_nat ) ) ).
% subringE(4)
thf(fact_890_subringE_I6_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H12: nat,H23: nat] :
( ( subrin2893992908230074586t_unit @ H2 @ R2 )
=> ( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subringE(6)
thf(fact_891_subringE_I3_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subrin2893992908230074586t_unit @ H2 @ R2 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ R2 ) @ H2 ) ) ).
% subringE(3)
thf(fact_892_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_893_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_894_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_895_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_896_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_897_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_898_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_899_subringE_I1_J,axiom,
! [H2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( subrin7689096310803670856t_unit @ H2 @ R2 )
=> ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% subringE(1)
thf(fact_900_subringE_I1_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subrin2893992908230074586t_unit @ H2 @ R2 )
=> ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% subringE(1)
thf(fact_901_ring_Ocarrier__is__subring,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( subrin7689096310803670856t_unit @ ( partia966996272515721803t_unit @ R2 ) @ R2 ) ) ).
% ring.carrier_is_subring
thf(fact_902_ring_Ocarrier__is__subring,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( subrin2893992908230074586t_unit @ ( partia3499330772048238685t_unit @ R2 ) @ R2 ) ) ).
% ring.carrier_is_subring
thf(fact_903_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_904_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_905_s_OsubdomainI,axiom,
! [H2: set_nat] :
( ( subcri1627753237249443161t_unit @ H2 @ ( mod_ring @ n ) )
=> ( ( ( one_na902338870878123981t_unit @ ( mod_ring @ n ) )
!= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
=> ( ! [H1: nat,H22: nat] :
( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ H1 @ H22 )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
=> ( ( H1
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
| ( H22
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ) ) ) )
=> ( subdom2148668005855505734t_unit @ H2 @ ( mod_ring @ n ) ) ) ) ) ).
% s.subdomainI
thf(fact_906_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_907_s_Osubfield__m__inv__simprule,axiom,
! [K2: set_nat,K: nat,A: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ K @ A ) @ K2 )
=> ( member_nat @ A @ K2 ) ) ) ) ) ).
% s.subfield_m_inv_simprule
thf(fact_908_compl__le__compl__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
= ( ord_less_eq_set_nat @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_909_s_Osubring__props_I2_J,axiom,
! [K2: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ K2 ) ) ).
% s.subring_props(2)
thf(fact_910_s_Osubring__props_I7_J,axiom,
! [K2: set_nat,H12: nat,H23: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( member_nat @ H12 @ K2 )
=> ( ( member_nat @ H23 @ K2 )
=> ( member_nat @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ H12 @ H23 ) @ K2 ) ) ) ) ).
% s.subring_props(7)
thf(fact_911_s_Osubring__props_I4_J,axiom,
! [K2: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( K2 != bot_bot_set_nat ) ) ).
% s.subring_props(4)
thf(fact_912_s_Osubring__props_I6_J,axiom,
! [K2: set_nat,H12: nat,H23: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( member_nat @ H12 @ K2 )
=> ( ( member_nat @ H23 @ K2 )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ H12 @ H23 ) @ K2 ) ) ) ) ).
% s.subring_props(6)
thf(fact_913_s_Osubring__props_I5_J,axiom,
! [K2: set_nat,H: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( member_nat @ H @ K2 )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ H ) @ K2 ) ) ) ).
% s.subring_props(5)
thf(fact_914_s_Osubring__props_I3_J,axiom,
! [K2: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( member_nat @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) @ K2 ) ) ).
% s.subring_props(3)
thf(fact_915_s_Osubring__props_I1_J,axiom,
! [K2: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.subring_props(1)
thf(fact_916_s_Oline__extension__smult__closed,axiom,
! [K2: set_nat,E: set_nat,A: nat,K: nat,U: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ! [K4: nat,V3: nat] :
( ( member_nat @ K4 @ K2 )
=> ( ( member_nat @ V3 @ E )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ K4 @ V3 ) @ E ) ) )
=> ( ( ord_less_eq_set_nat @ E @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ K @ K2 )
=> ( ( member_nat @ U @ ( embedd838748496991043025t_unit @ ( mod_ring @ n ) @ K2 @ A @ E ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ K @ U ) @ ( embedd838748496991043025t_unit @ ( mod_ring @ n ) @ K2 @ A @ E ) ) ) ) ) ) ) ) ).
% s.line_extension_smult_closed
thf(fact_917_subfieldE_I4_J,axiom,
! [K2: set_nat,R2: partia4692342223508353374t_unit,K1: nat,K22: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ K1 @ K2 )
=> ( ( member_nat @ K22 @ K2 )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ K1 @ K22 )
= ( mult_n6028127365542633569t_unit @ R2 @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_918_subdomainE_I2_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ H2 ) ) ).
% subdomainE(2)
thf(fact_919_subdomainE_I7_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H12: nat,H23: nat] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subdomainE(7)
thf(fact_920_subdomainE_I4_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( H2 != bot_bot_set_nat ) ) ).
% subdomainE(4)
thf(fact_921_subdomainE_I8_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H12: nat,H23: nat] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ H12 @ H23 )
= ( mult_n6028127365542633569t_unit @ R2 @ H23 @ H12 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_922_subdomainE_I6_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H12: nat,H23: nat] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ H12 @ H23 ) @ H2 ) ) ) ) ).
% subdomainE(6)
thf(fact_923_subdomainE_I3_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ R2 ) @ H2 ) ) ).
% subdomainE(3)
thf(fact_924_subfieldE_I3_J,axiom,
! [K2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ord_le4403425263959731960et_int @ K2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% subfieldE(3)
thf(fact_925_subfieldE_I3_J,axiom,
! [K2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% subfieldE(3)
thf(fact_926_subfieldE_I5_J,axiom,
! [K2: set_nat,R2: partia4692342223508353374t_unit,K1: nat,K22: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ K1 @ K2 )
=> ( ( member_nat @ K22 @ K2 )
=> ( ( ( mult_n6028127365542633569t_unit @ R2 @ K1 @ K22 )
= ( zero_n5149899317435570679t_unit @ R2 ) )
=> ( ( K1
= ( zero_n5149899317435570679t_unit @ R2 ) )
| ( K22
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_927_ring_Osubring__props_I2_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ K2 ) ) ) ).
% ring.subring_props(2)
thf(fact_928_ring_Osubring__props_I2_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ K2 ) ) ) ).
% ring.subring_props(2)
thf(fact_929_ring_Osubring__props_I7_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,H12: set_int,H23: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( member_set_int @ H12 @ K2 )
=> ( ( member_set_int @ H23 @ K2 )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R2 @ H12 @ H23 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(7)
thf(fact_930_ring_Osubring__props_I7_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,H12: nat,H23: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ H12 @ K2 )
=> ( ( member_nat @ H23 @ K2 )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ H12 @ H23 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(7)
thf(fact_931_ring_Osubring__props_I4_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( K2 != bot_bot_set_set_int ) ) ) ).
% ring.subring_props(4)
thf(fact_932_ring_Osubring__props_I4_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( K2 != bot_bot_set_nat ) ) ) ).
% ring.subring_props(4)
thf(fact_933_subfieldE_I6_J,axiom,
! [K2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( one_na902338870878123981t_unit @ R2 )
!= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ).
% subfieldE(6)
thf(fact_934_ring_Osubring__props_I6_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,H12: set_int,H23: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( member_set_int @ H12 @ K2 )
=> ( ( member_set_int @ H23 @ K2 )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ H12 @ H23 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(6)
thf(fact_935_ring_Osubring__props_I6_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,H12: nat,H23: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ H12 @ K2 )
=> ( ( member_nat @ H23 @ K2 )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ H12 @ H23 ) @ K2 ) ) ) ) ) ).
% ring.subring_props(6)
thf(fact_936_ring_Osubring__props_I3_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ R2 ) @ K2 ) ) ) ).
% ring.subring_props(3)
thf(fact_937_ring_Osubring__props_I3_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ R2 ) @ K2 ) ) ) ).
% ring.subring_props(3)
thf(fact_938_ring_Osubring__props_I5_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,H: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( member_set_int @ H @ K2 )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R2 @ H ) @ K2 ) ) ) ) ).
% ring.subring_props(5)
thf(fact_939_ring_Osubring__props_I5_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,H: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ H @ K2 )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ R2 @ H ) @ K2 ) ) ) ) ).
% ring.subring_props(5)
thf(fact_940_subdomainE_I1_J,axiom,
! [H2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( subdom1520866149873910708t_unit @ H2 @ R2 )
=> ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% subdomainE(1)
thf(fact_941_subdomainE_I1_J,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% subdomainE(1)
thf(fact_942_ring_Osubring__props_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ord_le4403425263959731960et_int @ K2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ).
% ring.subring_props(1)
thf(fact_943_ring_Osubring__props_I1_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ).
% ring.subring_props(1)
thf(fact_944_subdomain_Osubintegral,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit,H12: nat,H23: nat] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( ( member_nat @ H12 @ H2 )
=> ( ( member_nat @ H23 @ H2 )
=> ( ( ( mult_n6028127365542633569t_unit @ R2 @ H12 @ H23 )
= ( zero_n5149899317435570679t_unit @ R2 ) )
=> ( ( H12
= ( zero_n5149899317435570679t_unit @ R2 ) )
| ( H23
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_945_subdomain_Osub__one__not__zero,axiom,
! [H2: set_nat,R2: partia4692342223508353374t_unit] :
( ( subdom2148668005855505734t_unit @ H2 @ R2 )
=> ( ( one_na902338870878123981t_unit @ R2 )
!= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_946_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_947_ring_Oline__extension__smult__closed,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,E: set_set_int,A: set_int,K: set_int,U: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ! [K4: set_int,V3: set_int] :
( ( member_set_int @ K4 @ K2 )
=> ( ( member_set_int @ V3 @ E )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ K4 @ V3 ) @ E ) ) )
=> ( ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ K @ K2 )
=> ( ( member_set_int @ U @ ( embedd4283282269743769663t_unit @ R2 @ K2 @ A @ E ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ K @ U ) @ ( embedd4283282269743769663t_unit @ R2 @ K2 @ A @ E ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_948_ring_Oline__extension__smult__closed,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,E: set_nat,A: nat,K: nat,U: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ! [K4: nat,V3: nat] :
( ( member_nat @ K4 @ K2 )
=> ( ( member_nat @ V3 @ E )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ K4 @ V3 ) @ E ) ) )
=> ( ( ord_less_eq_set_nat @ E @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ K @ K2 )
=> ( ( member_nat @ U @ ( embedd838748496991043025t_unit @ R2 @ K2 @ A @ E ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ K @ U ) @ ( embedd838748496991043025t_unit @ R2 @ K2 @ A @ E ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_949_ring_Osubfield__m__inv__simprule,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,K: set_int,A: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( member_set_int @ K @ ( minus_8897228262479074673et_int @ K2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ K @ A ) @ K2 )
=> ( member_set_int @ A @ K2 ) ) ) ) ) ) ).
% ring.subfield_m_inv_simprule
thf(fact_950_ring_Osubfield__m__inv__simprule,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,K: nat,A: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ K @ A ) @ K2 )
=> ( member_nat @ A @ K2 ) ) ) ) ) ) ).
% ring.subfield_m_inv_simprule
thf(fact_951_compl__mono,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).
% compl_mono
thf(fact_952_compl__le__swap1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
=> ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% compl_le_swap1
thf(fact_953_compl__le__swap2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_954_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_955_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_956_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z3: int] :
? [N4: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_957_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_958_ring_OsubdomainI,axiom,
! [R2: partia4934656038542163276t_unit,H2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subcri1024317279029940167t_unit @ H2 @ R2 )
=> ( ( ( one_se8065767436706823081t_unit @ R2 )
!= ( zero_s6269048424454532197t_unit @ R2 ) )
=> ( ! [H1: set_int,H22: set_int] :
( ( member_set_int @ H1 @ H2 )
=> ( ( member_set_int @ H22 @ H2 )
=> ( ( ( mult_s3864001451298473021t_unit @ R2 @ H1 @ H22 )
= ( zero_s6269048424454532197t_unit @ R2 ) )
=> ( ( H1
= ( zero_s6269048424454532197t_unit @ R2 ) )
| ( H22
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ) )
=> ( subdom1520866149873910708t_unit @ H2 @ R2 ) ) ) ) ) ).
% ring.subdomainI
thf(fact_959_ring_OsubdomainI,axiom,
! [R2: partia4692342223508353374t_unit,H2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subcri1627753237249443161t_unit @ H2 @ R2 )
=> ( ( ( one_na902338870878123981t_unit @ R2 )
!= ( zero_n5149899317435570679t_unit @ R2 ) )
=> ( ! [H1: nat,H22: nat] :
( ( member_nat @ H1 @ H2 )
=> ( ( member_nat @ H22 @ H2 )
=> ( ( ( mult_n6028127365542633569t_unit @ R2 @ H1 @ H22 )
= ( zero_n5149899317435570679t_unit @ R2 ) )
=> ( ( H1
= ( zero_n5149899317435570679t_unit @ R2 ) )
| ( H22
= ( zero_n5149899317435570679t_unit @ R2 ) ) ) ) ) )
=> ( subdom2148668005855505734t_unit @ H2 @ R2 ) ) ) ) ) ).
% ring.subdomainI
thf(fact_960_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_961_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_962_int__cases2,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_963_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N3: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_964_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_965_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_966_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_967_int__cases,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_968_s_Osubfield__m__inv_I2_J,axiom,
! [K2: set_nat,K: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ K @ ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ K ) )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.subfield_m_inv(2)
thf(fact_969_s_Osubfield__m__inv_I3_J,axiom,
! [K2: set_nat,K: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ K ) @ K )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.subfield_m_inv(3)
thf(fact_970_s_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K2: set_nat,E: set_nat,V: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ E )
=> ( ( embedd2609395410403458802t_unit @ K2 @ V @ ( mod_ring @ n ) )
=> ( ( ord_less_eq_set_nat @ V @ E )
=> ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ V ) ) ) ) ) ).
% s.subalbegra_incl_imp_finite_dimension
thf(fact_971_s_Otelescopic__base__dim_I1_J,axiom,
! [K2: set_nat,F2: set_nat,E: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( subfie4892355163478727762t_unit @ F2 @ ( mod_ring @ n ) )
=> ( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ F2 )
=> ( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ F2 @ E )
=> ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ E ) ) ) ) ) ).
% s.telescopic_base_dim(1)
thf(fact_972_s_Oinv__eq__imp__eq,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ X )
= ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ Y ) )
=> ( X = Y ) ) ) ) ).
% s.inv_eq_imp_eq
thf(fact_973_s_Oinv__eq__one__eq,axiom,
! [X: nat] :
( ( member_nat @ X @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ X )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
= ( X
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.inv_eq_one_eq
thf(fact_974_s_Osum__space__dim_I1_J,axiom,
! [K2: set_nat,E: set_nat,F2: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ E )
=> ( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ F2 )
=> ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ E @ F2 ) ) ) ) ) ).
% s.sum_space_dim(1)
thf(fact_975_s_Ofinite__dimension__imp__subalgebra,axiom,
! [K2: set_nat,E: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ E )
=> ( embedd2609395410403458802t_unit @ K2 @ E @ ( mod_ring @ n ) ) ) ) ).
% s.finite_dimension_imp_subalgebra
thf(fact_976_s_Oinv__char,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Y @ X )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
=> ( ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ X )
= Y ) ) ) ) ) ).
% s.inv_char
thf(fact_977_s_Oinv__unique_H,axiom,
! [X: nat,Y: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ Y )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ Y @ X )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
=> ( Y
= ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ X ) ) ) ) ) ) ).
% s.inv_unique'
thf(fact_978_s_Oinv__eq__neg__one__eq,axiom,
! [X: nat] :
( ( member_nat @ X @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ X )
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) )
= ( X
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) ) ) ).
% s.inv_eq_neg_one_eq
thf(fact_979_s_Osubfield__m__inv_I1_J,axiom,
! [K2: set_nat,K: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ K ) @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) ) ) ) ).
% s.subfield_m_inv(1)
thf(fact_980_group_Oinv__inv,axiom,
! [G2: partia6999121085262744519t_unit,X: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( m_inv_8778965434624441924t_unit @ G2 @ ( m_inv_8778965434624441924t_unit @ G2 @ X ) )
= X ) ) ) ).
% group.inv_inv
thf(fact_981_group_Oinv__inv,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( m_inv_4894562657074299959t_unit @ G2 @ ( m_inv_4894562657074299959t_unit @ G2 @ X ) )
= X ) ) ) ).
% group.inv_inv
thf(fact_982_group_Oinv__inv,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( m_inv_3931797133766013019t_unit @ G2 @ ( m_inv_3931797133766013019t_unit @ G2 @ X ) )
= X ) ) ) ).
% group.inv_inv
thf(fact_983_s_Oinv__one,axiom,
( ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ).
% s.inv_one
thf(fact_984_s_OUnits__inv__Units,axiom,
! [X: nat] :
( ( member_nat @ X @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ X ) @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.Units_inv_Units
thf(fact_985_s_OUnits__inv__inv,axiom,
! [X: nat] :
( ( member_nat @ X @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ X ) )
= X ) ) ).
% s.Units_inv_inv
thf(fact_986_s_OUnits__inv__closed,axiom,
! [X: nat] :
( ( member_nat @ X @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ X ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.Units_inv_closed
thf(fact_987_s_Oinv__neg__one,axiom,
( ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) )
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.inv_neg_one
thf(fact_988_s_OUnits__l__inv,axiom,
! [X: nat] :
( ( member_nat @ X @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ X ) @ X )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.Units_l_inv
thf(fact_989_s_OUnits__r__inv,axiom,
! [X: nat] :
( ( member_nat @ X @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ X @ ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ X ) )
= ( one_na902338870878123981t_unit @ ( mod_ring @ n ) ) ) ) ).
% s.Units_r_inv
thf(fact_990_ring_Ofinite__dimension_Ocong,axiom,
embedd6096501799845681477t_unit = embedd6096501799845681477t_unit ).
% ring.finite_dimension.cong
thf(fact_991_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_992_monoid_Ounits__of__inv,axiom,
! [G2: partia6999121085262744519t_unit,X: nat] :
( ( monoid5435825805542372298t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1133518744711867743t_unit @ G2 ) )
=> ( ( m_inv_8778965434624441924t_unit @ ( units_1803606842677000328t_unit @ G2 ) @ X )
= ( m_inv_8778965434624441924t_unit @ G2 @ X ) ) ) ) ).
% monoid.units_of_inv
thf(fact_993_monoid_Ounits__of__inv,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( m_inv_8778965434624441924t_unit @ ( units_2222427131560811679t_unit @ G2 ) @ X )
= ( m_inv_3931797133766013019t_unit @ G2 @ X ) ) ) ) ).
% monoid.units_of_inv
thf(fact_994_group_Oinv__closed,axiom,
! [G2: partia6999121085262744519t_unit,X: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( member_nat @ ( m_inv_8778965434624441924t_unit @ G2 @ X ) @ ( partia5989097197943085638t_unit @ G2 ) ) ) ) ).
% group.inv_closed
thf(fact_995_group_Oinv__closed,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( member_set_int @ ( m_inv_4894562657074299959t_unit @ G2 @ X ) @ ( partia966996272515721803t_unit @ G2 ) ) ) ) ).
% group.inv_closed
thf(fact_996_group_Oinv__closed,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( member_nat @ ( m_inv_3931797133766013019t_unit @ G2 @ X ) @ ( partia3499330772048238685t_unit @ G2 ) ) ) ) ).
% group.inv_closed
thf(fact_997_monoid_Oinv__one,axiom,
! [G2: partia6999121085262744519t_unit] :
( ( monoid5435825805542372298t_unit @ G2 )
=> ( ( m_inv_8778965434624441924t_unit @ G2 @ ( one_nat_Product_unit @ G2 ) )
= ( one_nat_Product_unit @ G2 ) ) ) ).
% monoid.inv_one
thf(fact_998_monoid_Oinv__one,axiom,
! [G2: partia4692342223508353374t_unit] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( m_inv_3931797133766013019t_unit @ G2 @ ( one_na902338870878123981t_unit @ G2 ) )
= ( one_na902338870878123981t_unit @ G2 ) ) ) ).
% monoid.inv_one
thf(fact_999_monoid_OUnits__inv__Units,axiom,
! [G2: partia6999121085262744519t_unit,X: nat] :
( ( monoid5435825805542372298t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1133518744711867743t_unit @ G2 ) )
=> ( member_nat @ ( m_inv_8778965434624441924t_unit @ G2 @ X ) @ ( units_1133518744711867743t_unit @ G2 ) ) ) ) ).
% monoid.Units_inv_Units
thf(fact_1000_monoid_OUnits__inv__Units,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( member_nat @ ( m_inv_3931797133766013019t_unit @ G2 @ X ) @ ( units_1295200668666280182t_unit @ G2 ) ) ) ) ).
% monoid.Units_inv_Units
thf(fact_1001_monoid_OUnits__inv__inv,axiom,
! [G2: partia6999121085262744519t_unit,X: nat] :
( ( monoid5435825805542372298t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1133518744711867743t_unit @ G2 ) )
=> ( ( m_inv_8778965434624441924t_unit @ G2 @ ( m_inv_8778965434624441924t_unit @ G2 @ X ) )
= X ) ) ) ).
% monoid.Units_inv_inv
thf(fact_1002_monoid_OUnits__inv__inv,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( m_inv_3931797133766013019t_unit @ G2 @ ( m_inv_3931797133766013019t_unit @ G2 @ X ) )
= X ) ) ) ).
% monoid.Units_inv_inv
thf(fact_1003_monoid_Oinv__eq__imp__eq,axiom,
! [G2: partia6999121085262744519t_unit,X: nat,Y: nat] :
( ( monoid5435825805542372298t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1133518744711867743t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( units_1133518744711867743t_unit @ G2 ) )
=> ( ( ( m_inv_8778965434624441924t_unit @ G2 @ X )
= ( m_inv_8778965434624441924t_unit @ G2 @ Y ) )
=> ( X = Y ) ) ) ) ) ).
% monoid.inv_eq_imp_eq
thf(fact_1004_monoid_Oinv__eq__imp__eq,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( ( m_inv_3931797133766013019t_unit @ G2 @ X )
= ( m_inv_3931797133766013019t_unit @ G2 @ Y ) )
=> ( X = Y ) ) ) ) ) ).
% monoid.inv_eq_imp_eq
thf(fact_1005_group_Oinv__mult__group,axiom,
! [G2: partia6999121085262744519t_unit,X: nat,Y: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( m_inv_8778965434624441924t_unit @ G2 @ ( mult_n7532461433716462410t_unit @ G2 @ X @ Y ) )
= ( mult_n7532461433716462410t_unit @ G2 @ ( m_inv_8778965434624441924t_unit @ G2 @ Y ) @ ( m_inv_8778965434624441924t_unit @ G2 @ X ) ) ) ) ) ) ).
% group.inv_mult_group
thf(fact_1006_group_Oinv__mult__group,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( m_inv_4894562657074299959t_unit @ G2 @ ( mult_s3864001451298473021t_unit @ G2 @ X @ Y ) )
= ( mult_s3864001451298473021t_unit @ G2 @ ( m_inv_4894562657074299959t_unit @ G2 @ Y ) @ ( m_inv_4894562657074299959t_unit @ G2 @ X ) ) ) ) ) ) ).
% group.inv_mult_group
thf(fact_1007_group_Oinv__mult__group,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( m_inv_3931797133766013019t_unit @ G2 @ ( mult_n6028127365542633569t_unit @ G2 @ X @ Y ) )
= ( mult_n6028127365542633569t_unit @ G2 @ ( m_inv_3931797133766013019t_unit @ G2 @ Y ) @ ( m_inv_3931797133766013019t_unit @ G2 @ X ) ) ) ) ) ) ).
% group.inv_mult_group
thf(fact_1008_group_Oinv__solve__left,axiom,
! [G2: partia6999121085262744519t_unit,A: nat,B: nat,C: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ A @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ C @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( A
= ( mult_n7532461433716462410t_unit @ G2 @ ( m_inv_8778965434624441924t_unit @ G2 @ B ) @ C ) )
= ( C
= ( mult_n7532461433716462410t_unit @ G2 @ B @ A ) ) ) ) ) ) ) ).
% group.inv_solve_left
thf(fact_1009_group_Oinv__solve__left,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int,C: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( A
= ( mult_s3864001451298473021t_unit @ G2 @ ( m_inv_4894562657074299959t_unit @ G2 @ B ) @ C ) )
= ( C
= ( mult_s3864001451298473021t_unit @ G2 @ B @ A ) ) ) ) ) ) ) ).
% group.inv_solve_left
thf(fact_1010_group_Oinv__solve__left,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat,C: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( A
= ( mult_n6028127365542633569t_unit @ G2 @ ( m_inv_3931797133766013019t_unit @ G2 @ B ) @ C ) )
= ( C
= ( mult_n6028127365542633569t_unit @ G2 @ B @ A ) ) ) ) ) ) ) ).
% group.inv_solve_left
thf(fact_1011_group_Oinv__solve__left_H,axiom,
! [G2: partia6999121085262744519t_unit,A: nat,B: nat,C: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ A @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ C @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( ( mult_n7532461433716462410t_unit @ G2 @ ( m_inv_8778965434624441924t_unit @ G2 @ B ) @ C )
= A )
= ( C
= ( mult_n7532461433716462410t_unit @ G2 @ B @ A ) ) ) ) ) ) ) ).
% group.inv_solve_left'
thf(fact_1012_group_Oinv__solve__left_H,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int,C: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( ( mult_s3864001451298473021t_unit @ G2 @ ( m_inv_4894562657074299959t_unit @ G2 @ B ) @ C )
= A )
= ( C
= ( mult_s3864001451298473021t_unit @ G2 @ B @ A ) ) ) ) ) ) ) ).
% group.inv_solve_left'
thf(fact_1013_group_Oinv__solve__left_H,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat,C: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ ( m_inv_3931797133766013019t_unit @ G2 @ B ) @ C )
= A )
= ( C
= ( mult_n6028127365542633569t_unit @ G2 @ B @ A ) ) ) ) ) ) ) ).
% group.inv_solve_left'
thf(fact_1014_group_Oinv__solve__right,axiom,
! [G2: partia6999121085262744519t_unit,A: nat,B: nat,C: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ A @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ C @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( A
= ( mult_n7532461433716462410t_unit @ G2 @ B @ ( m_inv_8778965434624441924t_unit @ G2 @ C ) ) )
= ( B
= ( mult_n7532461433716462410t_unit @ G2 @ A @ C ) ) ) ) ) ) ) ).
% group.inv_solve_right
thf(fact_1015_group_Oinv__solve__right,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int,C: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( A
= ( mult_s3864001451298473021t_unit @ G2 @ B @ ( m_inv_4894562657074299959t_unit @ G2 @ C ) ) )
= ( B
= ( mult_s3864001451298473021t_unit @ G2 @ A @ C ) ) ) ) ) ) ) ).
% group.inv_solve_right
thf(fact_1016_group_Oinv__solve__right,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat,C: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( A
= ( mult_n6028127365542633569t_unit @ G2 @ B @ ( m_inv_3931797133766013019t_unit @ G2 @ C ) ) )
= ( B
= ( mult_n6028127365542633569t_unit @ G2 @ A @ C ) ) ) ) ) ) ) ).
% group.inv_solve_right
thf(fact_1017_group_Oinv__solve__right_H,axiom,
! [G2: partia6999121085262744519t_unit,A: nat,B: nat,C: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ A @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ C @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( ( mult_n7532461433716462410t_unit @ G2 @ B @ ( m_inv_8778965434624441924t_unit @ G2 @ C ) )
= A )
= ( B
= ( mult_n7532461433716462410t_unit @ G2 @ A @ C ) ) ) ) ) ) ) ).
% group.inv_solve_right'
thf(fact_1018_group_Oinv__solve__right_H,axiom,
! [G2: partia4934656038542163276t_unit,A: set_int,B: set_int,C: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( ( mult_s3864001451298473021t_unit @ G2 @ B @ ( m_inv_4894562657074299959t_unit @ G2 @ C ) )
= A )
= ( B
= ( mult_s3864001451298473021t_unit @ G2 @ A @ C ) ) ) ) ) ) ) ).
% group.inv_solve_right'
thf(fact_1019_group_Oinv__solve__right_H,axiom,
! [G2: partia4692342223508353374t_unit,A: nat,B: nat,C: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ B @ ( m_inv_3931797133766013019t_unit @ G2 @ C ) )
= A )
= ( B
= ( mult_n6028127365542633569t_unit @ G2 @ A @ C ) ) ) ) ) ) ) ).
% group.inv_solve_right'
thf(fact_1020_group_Oinv__eq__1__iff,axiom,
! [G2: partia6999121085262744519t_unit,X: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( ( m_inv_8778965434624441924t_unit @ G2 @ X )
= ( one_nat_Product_unit @ G2 ) )
= ( X
= ( one_nat_Product_unit @ G2 ) ) ) ) ) ).
% group.inv_eq_1_iff
thf(fact_1021_group_Oinv__eq__1__iff,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( ( m_inv_4894562657074299959t_unit @ G2 @ X )
= ( one_se8065767436706823081t_unit @ G2 ) )
= ( X
= ( one_se8065767436706823081t_unit @ G2 ) ) ) ) ) ).
% group.inv_eq_1_iff
thf(fact_1022_group_Oinv__eq__1__iff,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( ( m_inv_3931797133766013019t_unit @ G2 @ X )
= ( one_na902338870878123981t_unit @ G2 ) )
= ( X
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ) ).
% group.inv_eq_1_iff
thf(fact_1023_monoid_OUnits__inv__closed,axiom,
! [G2: partia6999121085262744519t_unit,X: nat] :
( ( monoid5435825805542372298t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1133518744711867743t_unit @ G2 ) )
=> ( member_nat @ ( m_inv_8778965434624441924t_unit @ G2 @ X ) @ ( partia5989097197943085638t_unit @ G2 ) ) ) ) ).
% monoid.Units_inv_closed
thf(fact_1024_monoid_OUnits__inv__closed,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ X @ ( units_4038138251425117394t_unit @ G2 ) )
=> ( member_set_int @ ( m_inv_4894562657074299959t_unit @ G2 @ X ) @ ( partia966996272515721803t_unit @ G2 ) ) ) ) ).
% monoid.Units_inv_closed
thf(fact_1025_monoid_OUnits__inv__closed,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( member_nat @ ( m_inv_3931797133766013019t_unit @ G2 @ X ) @ ( partia3499330772048238685t_unit @ G2 ) ) ) ) ).
% monoid.Units_inv_closed
thf(fact_1026_monoid_Oinv__eq__one__eq,axiom,
! [G2: partia6999121085262744519t_unit,X: nat] :
( ( monoid5435825805542372298t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1133518744711867743t_unit @ G2 ) )
=> ( ( ( m_inv_8778965434624441924t_unit @ G2 @ X )
= ( one_nat_Product_unit @ G2 ) )
= ( X
= ( one_nat_Product_unit @ G2 ) ) ) ) ) ).
% monoid.inv_eq_one_eq
thf(fact_1027_monoid_Oinv__eq__one__eq,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( ( m_inv_3931797133766013019t_unit @ G2 @ X )
= ( one_na902338870878123981t_unit @ G2 ) )
= ( X
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ) ).
% monoid.inv_eq_one_eq
thf(fact_1028_ring_Oinv__neg__one,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( m_inv_4894562657074299959t_unit @ R2 @ ( a_inv_5951419416477254493t_unit @ R2 @ ( one_se8065767436706823081t_unit @ R2 ) ) )
= ( a_inv_5951419416477254493t_unit @ R2 @ ( one_se8065767436706823081t_unit @ R2 ) ) ) ) ).
% ring.inv_neg_one
thf(fact_1029_ring_Oinv__neg__one,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( m_inv_3931797133766013019t_unit @ R2 @ ( a_inv_2472168910397739247t_unit @ R2 @ ( one_na902338870878123981t_unit @ R2 ) ) )
= ( a_inv_2472168910397739247t_unit @ R2 @ ( one_na902338870878123981t_unit @ R2 ) ) ) ) ).
% ring.inv_neg_one
thf(fact_1030_ring_Otelescopic__base__dim_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,F2: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( subfie3888952257595785920t_unit @ F2 @ R2 )
=> ( ( embedd8246663962306818995t_unit @ R2 @ K2 @ F2 )
=> ( ( embedd8246663962306818995t_unit @ R2 @ F2 @ E )
=> ( embedd8246663962306818995t_unit @ R2 @ K2 @ E ) ) ) ) ) ) ).
% ring.telescopic_base_dim(1)
thf(fact_1031_ring_Otelescopic__base__dim_I1_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,F2: set_nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( subfie4892355163478727762t_unit @ F2 @ R2 )
=> ( ( embedd6096501799845681477t_unit @ R2 @ K2 @ F2 )
=> ( ( embedd6096501799845681477t_unit @ R2 @ F2 @ E )
=> ( embedd6096501799845681477t_unit @ R2 @ K2 @ E ) ) ) ) ) ) ).
% ring.telescopic_base_dim(1)
thf(fact_1032_group_Oinv__equality,axiom,
! [G2: partia6999121085262744519t_unit,Y: nat,X: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( ( mult_n7532461433716462410t_unit @ G2 @ Y @ X )
= ( one_nat_Product_unit @ G2 ) )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( m_inv_8778965434624441924t_unit @ G2 @ X )
= Y ) ) ) ) ) ).
% group.inv_equality
thf(fact_1033_group_Oinv__equality,axiom,
! [G2: partia4934656038542163276t_unit,Y: set_int,X: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( ( mult_s3864001451298473021t_unit @ G2 @ Y @ X )
= ( one_se8065767436706823081t_unit @ G2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( m_inv_4894562657074299959t_unit @ G2 @ X )
= Y ) ) ) ) ) ).
% group.inv_equality
thf(fact_1034_group_Oinv__equality,axiom,
! [G2: partia4692342223508353374t_unit,Y: nat,X: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ Y @ X )
= ( one_na902338870878123981t_unit @ G2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( m_inv_3931797133766013019t_unit @ G2 @ X )
= Y ) ) ) ) ) ).
% group.inv_equality
thf(fact_1035_group_Or__inv,axiom,
! [G2: partia6999121085262744519t_unit,X: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( mult_n7532461433716462410t_unit @ G2 @ X @ ( m_inv_8778965434624441924t_unit @ G2 @ X ) )
= ( one_nat_Product_unit @ G2 ) ) ) ) ).
% group.r_inv
thf(fact_1036_group_Or__inv,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( mult_s3864001451298473021t_unit @ G2 @ X @ ( m_inv_4894562657074299959t_unit @ G2 @ X ) )
= ( one_se8065767436706823081t_unit @ G2 ) ) ) ) ).
% group.r_inv
thf(fact_1037_group_Or__inv,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ X @ ( m_inv_3931797133766013019t_unit @ G2 @ X ) )
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ).
% group.r_inv
thf(fact_1038_group_Ol__inv,axiom,
! [G2: partia6999121085262744519t_unit,X: nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( mult_n7532461433716462410t_unit @ G2 @ ( m_inv_8778965434624441924t_unit @ G2 @ X ) @ X )
= ( one_nat_Product_unit @ G2 ) ) ) ) ).
% group.l_inv
thf(fact_1039_group_Ol__inv,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( mult_s3864001451298473021t_unit @ G2 @ ( m_inv_4894562657074299959t_unit @ G2 @ X ) @ X )
= ( one_se8065767436706823081t_unit @ G2 ) ) ) ) ).
% group.l_inv
thf(fact_1040_group_Ol__inv,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ ( m_inv_3931797133766013019t_unit @ G2 @ X ) @ X )
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ).
% group.l_inv
thf(fact_1041_monoid_Oinv__char,axiom,
! [G2: partia6999121085262744519t_unit,X: nat,Y: nat] :
( ( monoid5435825805542372298t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( ( mult_n7532461433716462410t_unit @ G2 @ X @ Y )
= ( one_nat_Product_unit @ G2 ) )
=> ( ( ( mult_n7532461433716462410t_unit @ G2 @ Y @ X )
= ( one_nat_Product_unit @ G2 ) )
=> ( ( m_inv_8778965434624441924t_unit @ G2 @ X )
= Y ) ) ) ) ) ) ).
% monoid.inv_char
thf(fact_1042_monoid_Oinv__char,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( ( mult_s3864001451298473021t_unit @ G2 @ X @ Y )
= ( one_se8065767436706823081t_unit @ G2 ) )
=> ( ( ( mult_s3864001451298473021t_unit @ G2 @ Y @ X )
= ( one_se8065767436706823081t_unit @ G2 ) )
=> ( ( m_inv_4894562657074299959t_unit @ G2 @ X )
= Y ) ) ) ) ) ) ).
% monoid.inv_char
thf(fact_1043_monoid_Oinv__char,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ X @ Y )
= ( one_na902338870878123981t_unit @ G2 ) )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ Y @ X )
= ( one_na902338870878123981t_unit @ G2 ) )
=> ( ( m_inv_3931797133766013019t_unit @ G2 @ X )
= Y ) ) ) ) ) ) ).
% monoid.inv_char
thf(fact_1044_monoid_Oinv__unique_H,axiom,
! [G2: partia6999121085262744519t_unit,X: nat,Y: nat] :
( ( monoid5435825805542372298t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( ( mult_n7532461433716462410t_unit @ G2 @ X @ Y )
= ( one_nat_Product_unit @ G2 ) )
=> ( ( ( mult_n7532461433716462410t_unit @ G2 @ Y @ X )
= ( one_nat_Product_unit @ G2 ) )
=> ( Y
= ( m_inv_8778965434624441924t_unit @ G2 @ X ) ) ) ) ) ) ) ).
% monoid.inv_unique'
thf(fact_1045_monoid_Oinv__unique_H,axiom,
! [G2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( monoid833175047693460669t_unit @ G2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( ( mult_s3864001451298473021t_unit @ G2 @ X @ Y )
= ( one_se8065767436706823081t_unit @ G2 ) )
=> ( ( ( mult_s3864001451298473021t_unit @ G2 @ Y @ X )
= ( one_se8065767436706823081t_unit @ G2 ) )
=> ( Y
= ( m_inv_4894562657074299959t_unit @ G2 @ X ) ) ) ) ) ) ) ).
% monoid.inv_unique'
thf(fact_1046_monoid_Oinv__unique_H,axiom,
! [G2: partia4692342223508353374t_unit,X: nat,Y: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ X @ Y )
= ( one_na902338870878123981t_unit @ G2 ) )
=> ( ( ( mult_n6028127365542633569t_unit @ G2 @ Y @ X )
= ( one_na902338870878123981t_unit @ G2 ) )
=> ( Y
= ( m_inv_3931797133766013019t_unit @ G2 @ X ) ) ) ) ) ) ) ).
% monoid.inv_unique'
thf(fact_1047_monoid_OUnits__r__inv,axiom,
! [G2: partia6999121085262744519t_unit,X: nat] :
( ( monoid5435825805542372298t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1133518744711867743t_unit @ G2 ) )
=> ( ( mult_n7532461433716462410t_unit @ G2 @ X @ ( m_inv_8778965434624441924t_unit @ G2 @ X ) )
= ( one_nat_Product_unit @ G2 ) ) ) ) ).
% monoid.Units_r_inv
thf(fact_1048_monoid_OUnits__r__inv,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ X @ ( m_inv_3931797133766013019t_unit @ G2 @ X ) )
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ).
% monoid.Units_r_inv
thf(fact_1049_monoid_OUnits__l__inv,axiom,
! [G2: partia6999121085262744519t_unit,X: nat] :
( ( monoid5435825805542372298t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1133518744711867743t_unit @ G2 ) )
=> ( ( mult_n7532461433716462410t_unit @ G2 @ ( m_inv_8778965434624441924t_unit @ G2 @ X ) @ X )
= ( one_nat_Product_unit @ G2 ) ) ) ) ).
% monoid.Units_l_inv
thf(fact_1050_monoid_OUnits__l__inv,axiom,
! [G2: partia4692342223508353374t_unit,X: nat] :
( ( monoid4477821099152670945t_unit @ G2 )
=> ( ( member_nat @ X @ ( units_1295200668666280182t_unit @ G2 ) )
=> ( ( mult_n6028127365542633569t_unit @ G2 @ ( m_inv_3931797133766013019t_unit @ G2 @ X ) @ X )
= ( one_na902338870878123981t_unit @ G2 ) ) ) ) ).
% monoid.Units_l_inv
thf(fact_1051_ring_Oinv__eq__neg__one__eq,axiom,
! [R2: partia4934656038542163276t_unit,X: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ X @ ( units_4038138251425117394t_unit @ R2 ) )
=> ( ( ( m_inv_4894562657074299959t_unit @ R2 @ X )
= ( a_inv_5951419416477254493t_unit @ R2 @ ( one_se8065767436706823081t_unit @ R2 ) ) )
= ( X
= ( a_inv_5951419416477254493t_unit @ R2 @ ( one_se8065767436706823081t_unit @ R2 ) ) ) ) ) ) ).
% ring.inv_eq_neg_one_eq
thf(fact_1052_ring_Oinv__eq__neg__one__eq,axiom,
! [R2: partia4692342223508353374t_unit,X: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ X @ ( units_1295200668666280182t_unit @ R2 ) )
=> ( ( ( m_inv_3931797133766013019t_unit @ R2 @ X )
= ( a_inv_2472168910397739247t_unit @ R2 @ ( one_na902338870878123981t_unit @ R2 ) ) )
= ( X
= ( a_inv_2472168910397739247t_unit @ R2 @ ( one_na902338870878123981t_unit @ R2 ) ) ) ) ) ) ).
% ring.inv_eq_neg_one_eq
thf(fact_1053_ring_Osum__space__dim_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,E: set_set_int,F2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd8246663962306818995t_unit @ R2 @ K2 @ E )
=> ( ( embedd8246663962306818995t_unit @ R2 @ K2 @ F2 )
=> ( embedd8246663962306818995t_unit @ R2 @ K2 @ ( set_ad273131178244904872t_unit @ R2 @ E @ F2 ) ) ) ) ) ) ).
% ring.sum_space_dim(1)
thf(fact_1054_ring_Osum__space__dim_I1_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,E: set_nat,F2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd6096501799845681477t_unit @ R2 @ K2 @ E )
=> ( ( embedd6096501799845681477t_unit @ R2 @ K2 @ F2 )
=> ( embedd6096501799845681477t_unit @ R2 @ K2 @ ( set_ad320919470248169786t_unit @ R2 @ E @ F2 ) ) ) ) ) ) ).
% ring.sum_space_dim(1)
thf(fact_1055_ring_Ofinite__dimension__imp__subalgebra,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd8246663962306818995t_unit @ R2 @ K2 @ E )
=> ( embedd2743979684206749024t_unit @ K2 @ E @ R2 ) ) ) ) ).
% ring.finite_dimension_imp_subalgebra
thf(fact_1056_ring_Ofinite__dimension__imp__subalgebra,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd6096501799845681477t_unit @ R2 @ K2 @ E )
=> ( embedd2609395410403458802t_unit @ K2 @ E @ R2 ) ) ) ) ).
% ring.finite_dimension_imp_subalgebra
thf(fact_1057_group_Oone__in__subset,axiom,
! [G2: partia6999121085262744519t_unit,H2: set_nat] :
( ( group_1007331108230521615t_unit @ G2 )
=> ( ( ord_less_eq_set_nat @ H2 @ ( partia5989097197943085638t_unit @ G2 ) )
=> ( ( H2 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ H2 )
=> ( member_nat @ ( m_inv_8778965434624441924t_unit @ G2 @ X4 ) @ H2 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ H2 )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ H2 )
=> ( member_nat @ ( mult_n7532461433716462410t_unit @ G2 @ X4 @ Xa2 ) @ H2 ) ) )
=> ( member_nat @ ( one_nat_Product_unit @ G2 ) @ H2 ) ) ) ) ) ) ).
% group.one_in_subset
thf(fact_1058_group_Oone__in__subset,axiom,
! [G2: partia4934656038542163276t_unit,H2: set_set_int] :
( ( group_1453825718996320898t_unit @ G2 )
=> ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( H2 != bot_bot_set_set_int )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ H2 )
=> ( member_set_int @ ( m_inv_4894562657074299959t_unit @ G2 @ X4 ) @ H2 ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ H2 )
=> ! [Xa2: set_int] :
( ( member_set_int @ Xa2 @ H2 )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ G2 @ X4 @ Xa2 ) @ H2 ) ) )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ G2 ) @ H2 ) ) ) ) ) ) ).
% group.one_in_subset
thf(fact_1059_group_Oone__in__subset,axiom,
! [G2: partia4692342223508353374t_unit,H2: set_nat] :
( ( group_2402336746480010918t_unit @ G2 )
=> ( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( H2 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ H2 )
=> ( member_nat @ ( m_inv_3931797133766013019t_unit @ G2 @ X4 ) @ H2 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ H2 )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ H2 )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ G2 @ X4 @ Xa2 ) @ H2 ) ) )
=> ( member_nat @ ( one_na902338870878123981t_unit @ G2 ) @ H2 ) ) ) ) ) ) ).
% group.one_in_subset
thf(fact_1060_ring_Osubfield__m__inv_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,K: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( member_set_int @ K @ ( minus_8897228262479074673et_int @ K2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) )
=> ( member_set_int @ ( m_inv_4894562657074299959t_unit @ R2 @ K ) @ ( minus_8897228262479074673et_int @ K2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) ) ) ) ) ).
% ring.subfield_m_inv(1)
thf(fact_1061_ring_Osubfield__m__inv_I1_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,K: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( m_inv_3931797133766013019t_unit @ R2 @ K ) @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) ) ) ) ) ).
% ring.subfield_m_inv(1)
thf(fact_1062_ring_Osubalbegra__incl__imp__finite__dimension,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,E: set_set_int,V: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd8246663962306818995t_unit @ R2 @ K2 @ E )
=> ( ( embedd2743979684206749024t_unit @ K2 @ V @ R2 )
=> ( ( ord_le4403425263959731960et_int @ V @ E )
=> ( embedd8246663962306818995t_unit @ R2 @ K2 @ V ) ) ) ) ) ) ).
% ring.subalbegra_incl_imp_finite_dimension
thf(fact_1063_ring_Osubalbegra__incl__imp__finite__dimension,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,E: set_nat,V: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd6096501799845681477t_unit @ R2 @ K2 @ E )
=> ( ( embedd2609395410403458802t_unit @ K2 @ V @ R2 )
=> ( ( ord_less_eq_set_nat @ V @ E )
=> ( embedd6096501799845681477t_unit @ R2 @ K2 @ V ) ) ) ) ) ) ).
% ring.subalbegra_incl_imp_finite_dimension
thf(fact_1064_ring_Osubfield__m__inv_I3_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,K: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( member_set_int @ K @ ( minus_8897228262479074673et_int @ K2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ ( m_inv_4894562657074299959t_unit @ R2 @ K ) @ K )
= ( one_se8065767436706823081t_unit @ R2 ) ) ) ) ) ).
% ring.subfield_m_inv(3)
thf(fact_1065_ring_Osubfield__m__inv_I3_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,K: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ ( m_inv_3931797133766013019t_unit @ R2 @ K ) @ K )
= ( one_na902338870878123981t_unit @ R2 ) ) ) ) ) ).
% ring.subfield_m_inv(3)
thf(fact_1066_ring_Osubfield__m__inv_I2_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,K: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( member_set_int @ K @ ( minus_8897228262479074673et_int @ K2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) )
=> ( ( mult_s3864001451298473021t_unit @ R2 @ K @ ( m_inv_4894562657074299959t_unit @ R2 @ K ) )
= ( one_se8065767436706823081t_unit @ R2 ) ) ) ) ) ).
% ring.subfield_m_inv(2)
thf(fact_1067_ring_Osubfield__m__inv_I2_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,K: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) )
=> ( ( mult_n6028127365542633569t_unit @ R2 @ K @ ( m_inv_3931797133766013019t_unit @ R2 @ K ) )
= ( one_na902338870878123981t_unit @ R2 ) ) ) ) ) ).
% ring.subfield_m_inv(2)
thf(fact_1068_s_Ospace__subgroup__props_I6_J,axiom,
! [K2: set_nat,N: nat,E: set_nat,K: nat,A: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ K @ A ) @ E )
=> ( member_nat @ A @ E ) ) ) ) ) ) ).
% s.space_subgroup_props(6)
thf(fact_1069_s_Oint__embed__mult__aux,axiom,
! [X: int,Y: nat] :
( ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ ( times_times_int @ X @ ( semiri1314217659103216013at_int @ Y ) ) )
= ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ X ) @ ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).
% s.int_embed_mult_aux
thf(fact_1070_s_Oa__rcos__assoc__lcos,axiom,
! [H2: set_nat,K2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ H2 @ X ) @ K2 )
= ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ H2 @ ( a_l_co3340896127261556338t_unit @ ( mod_ring @ n ) @ X @ K2 ) ) ) ) ) ) ).
% s.a_rcos_assoc_lcos
thf(fact_1071_s_Oint__embed__closed,axiom,
! [K: int] : ( member_nat @ ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ K ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ).
% s.int_embed_closed
thf(fact_1072_s_Odimension__is__inj,axiom,
! [K2: set_nat,N: nat,E: set_nat,M: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ M @ K2 @ E )
=> ( N = M ) ) ) ) ).
% s.dimension_is_inj
thf(fact_1073_s_Oint__embed__range,axiom,
! [K2: set_nat,K: int] :
( ( subrin2893992908230074586t_unit @ K2 @ ( mod_ring @ n ) )
=> ( member_nat @ ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ K ) @ K2 ) ) ).
% s.int_embed_range
thf(fact_1074_s_Ofinite__dimensionE_H,axiom,
! [K2: set_nat,E: set_nat] :
( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ E )
=> ~ ! [N3: nat] :
~ ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N3 @ K2 @ E ) ) ).
% s.finite_dimensionE'
thf(fact_1075_s_Ofinite__dimensionI,axiom,
! [N: nat,K2: set_nat,E: set_nat] :
( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ E ) ) ).
% s.finite_dimensionI
thf(fact_1076_s_Ofinite__dimension__def,axiom,
! [K2: set_nat,E: set_nat] :
( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ E )
= ( ? [N4: nat] : ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N4 @ K2 @ E ) ) ) ).
% s.finite_dimension_def
thf(fact_1077_s_Oint__embed__add,axiom,
! [X: int,Y: int] :
( ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ ( plus_plus_int @ X @ Y ) )
= ( add_nat_Product_unit @ ( mod_ring @ n ) @ ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ X ) @ ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ Y ) ) ) ).
% s.int_embed_add
thf(fact_1078_s_Oa__r__coset__subset__G,axiom,
! [H2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ord_less_eq_set_nat @ ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ H2 @ X ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.a_r_coset_subset_G
thf(fact_1079_s_Ospace__subgroup__props_I2_J,axiom,
! [K2: set_nat,N: nat,E: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ E ) ) ) ).
% s.space_subgroup_props(2)
thf(fact_1080_s_Ospace__subgroup__props_I3_J,axiom,
! [K2: set_nat,N: nat,E: set_nat,V1: nat,V22: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( ( member_nat @ V1 @ E )
=> ( ( member_nat @ V22 @ E )
=> ( member_nat @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ V1 @ V22 ) @ E ) ) ) ) ) ).
% s.space_subgroup_props(3)
thf(fact_1081_s_Otelescopic__base,axiom,
! [K2: set_nat,F2: set_nat,N: nat,M: nat,E: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( subfie4892355163478727762t_unit @ F2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ F2 )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ M @ F2 @ E )
=> ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ ( times_times_nat @ N @ M ) @ K2 @ E ) ) ) ) ) ).
% s.telescopic_base
thf(fact_1082_s_Oint__embed__mult,axiom,
! [X: int,Y: int] :
( ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ ( times_times_int @ X @ Y ) )
= ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ X ) @ ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ Y ) ) ) ).
% s.int_embed_mult
thf(fact_1083_s_Ospace__subgroup__props_I5_J,axiom,
! [K2: set_nat,N: nat,E: set_nat,K: nat,V2: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( ( member_nat @ K @ K2 )
=> ( ( member_nat @ V2 @ E )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ K @ V2 ) @ E ) ) ) ) ) ).
% s.space_subgroup_props(5)
thf(fact_1084_s_Oint__embed__inv,axiom,
! [X: int] :
( ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ ( uminus_uminus_int @ X ) )
= ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ X ) ) ) ).
% s.int_embed_inv
thf(fact_1085_s_Ospace__subgroup__props_I4_J,axiom,
! [K2: set_nat,N: nat,E: set_nat,V2: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( ( member_nat @ V2 @ E )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ V2 ) @ E ) ) ) ) ).
% s.space_subgroup_props(4)
thf(fact_1086_s_Ounique__dimension,axiom,
! [K2: set_nat,E: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd6096501799845681477t_unit @ ( mod_ring @ n ) @ K2 @ E )
=> ? [X4: nat] :
( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ X4 @ K2 @ E )
& ! [Y5: nat] :
( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ Y5 @ K2 @ E )
=> ( Y5 = X4 ) ) ) ) ) ).
% s.unique_dimension
thf(fact_1087_s_Oint__embed__diff,axiom,
! [X: int,Y: int] :
( ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ ( minus_minus_int @ X @ Y ) )
= ( a_minu1751788497103602224t_unit @ ( mod_ring @ n ) @ ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ X ) @ ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ Y ) ) ) ).
% s.int_embed_diff
thf(fact_1088_s_Oa__rcosI,axiom,
! [H: nat,H2: set_nat,X: nat] :
( ( member_nat @ H @ H2 )
=> ( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_nat @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ H @ X ) @ ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ H2 @ X ) ) ) ) ) ).
% s.a_rcosI
thf(fact_1089_s_Oa__coset__add__assoc,axiom,
! [M2: set_nat,G: nat,H: nat] :
( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ G @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ H @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ G ) @ H )
= ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ G @ H ) ) ) ) ) ) ).
% s.a_coset_add_assoc
thf(fact_1090_s_Ospace__subgroup__props_I1_J,axiom,
! [K2: set_nat,N: nat,E: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( ord_less_eq_set_nat @ E @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.space_subgroup_props(1)
thf(fact_1091_s_Ounits__of__inv,axiom,
! [X: nat] :
( ( member_nat @ X @ ( units_1295200668666280182t_unit @ ( mod_ring @ n ) ) )
=> ( ( m_inv_8778965434624441924t_unit @ ( units_2222427131560811679t_unit @ ( mod_ring @ n ) ) @ X )
= ( m_inv_3931797133766013019t_unit @ ( mod_ring @ n ) @ X ) ) ) ).
% s.units_of_inv
thf(fact_1092_s_OSuc__dim,axiom,
! [V2: nat,E: set_nat,N: nat,K2: set_nat] :
( ( member_nat @ V2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ~ ( member_nat @ V2 @ E )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ ( suc @ N ) @ K2 @ ( embedd838748496991043025t_unit @ ( mod_ring @ n ) @ K2 @ V2 @ E ) ) ) ) ) ).
% s.Suc_dim
thf(fact_1093_s_Oa__setmult__rcos__assoc,axiom,
! [H2: set_nat,K2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ H2 @ ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ K2 @ X ) )
= ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ H2 @ K2 ) @ X ) ) ) ) ) ).
% s.a_setmult_rcos_assoc
thf(fact_1094_s_Oa__coset__add__inv2,axiom,
! [M2: set_nat,X: nat,Y: nat] :
( ( ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ X )
= ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ Y ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ Y ) ) )
= M2 ) ) ) ) ) ).
% s.a_coset_add_inv2
thf(fact_1095_s_Oa__coset__add__inv1,axiom,
! [M2: set_nat,X: nat,Y: nat] :
( ( ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ ( add_nat_Product_unit @ ( mod_ring @ n ) @ X @ ( a_inv_2472168910397739247t_unit @ ( mod_ring @ n ) @ Y ) ) )
= M2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ X )
= ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ Y ) ) ) ) ) ) ).
% s.a_coset_add_inv1
thf(fact_1096_s_Odimension__backwards,axiom,
! [K2: set_nat,N: nat,E: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ ( suc @ N ) @ K2 @ E )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
& ? [E3: set_nat] :
( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E3 )
& ~ ( member_nat @ X4 @ E3 )
& ( E
= ( embedd838748496991043025t_unit @ ( mod_ring @ n ) @ K2 @ X4 @ E3 ) ) ) ) ) ) ).
% s.dimension_backwards
thf(fact_1097_s_Oa__coset__add__zero,axiom,
! [M2: set_nat] :
( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ M2 @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
= M2 ) ) ).
% s.a_coset_add_zero
thf(fact_1098_ring_Odimension_Ocong,axiom,
embedd5688180257602933782t_unit = embedd5688180257602933782t_unit ).
% ring.dimension.cong
thf(fact_1099_ring_Odimension__is__inj,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int,M: nat] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( ( embedd646006463343340164t_unit @ R2 @ M @ K2 @ E )
=> ( N = M ) ) ) ) ) ).
% ring.dimension_is_inj
thf(fact_1100_ring_Odimension__is__inj,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat,M: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( ( embedd5688180257602933782t_unit @ R2 @ M @ K2 @ E )
=> ( N = M ) ) ) ) ) ).
% ring.dimension_is_inj
thf(fact_1101_ring_Ofinite__dimension__def,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( embedd8246663962306818995t_unit @ R2 @ K2 @ E )
= ( ? [N4: nat] : ( embedd646006463343340164t_unit @ R2 @ N4 @ K2 @ E ) ) ) ) ).
% ring.finite_dimension_def
thf(fact_1102_ring_Ofinite__dimension__def,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( embedd6096501799845681477t_unit @ R2 @ K2 @ E )
= ( ? [N4: nat] : ( embedd5688180257602933782t_unit @ R2 @ N4 @ K2 @ E ) ) ) ) ).
% ring.finite_dimension_def
thf(fact_1103_ring_Ofinite__dimensionE_H,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( embedd8246663962306818995t_unit @ R2 @ K2 @ E )
=> ~ ! [N3: nat] :
~ ( embedd646006463343340164t_unit @ R2 @ N3 @ K2 @ E ) ) ) ).
% ring.finite_dimensionE'
thf(fact_1104_ring_Ofinite__dimensionE_H,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( embedd6096501799845681477t_unit @ R2 @ K2 @ E )
=> ~ ! [N3: nat] :
~ ( embedd5688180257602933782t_unit @ R2 @ N3 @ K2 @ E ) ) ) ).
% ring.finite_dimensionE'
thf(fact_1105_ring_Ofinite__dimensionI,axiom,
! [R2: partia4934656038542163276t_unit,N: nat,K2: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( embedd8246663962306818995t_unit @ R2 @ K2 @ E ) ) ) ).
% ring.finite_dimensionI
thf(fact_1106_ring_Ofinite__dimensionI,axiom,
! [R2: partia4692342223508353374t_unit,N: nat,K2: set_nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( embedd6096501799845681477t_unit @ R2 @ K2 @ E ) ) ) ).
% ring.finite_dimensionI
thf(fact_1107_abelian__monoid_Oa__r__coset__subset__G,axiom,
! [G2: partia4934656038542163276t_unit,H2: set_set_int,X: set_int] :
( ( abelia3815030880812984441t_unit @ G2 )
=> ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ord_le4403425263959731960et_int @ ( a_r_co692709266861932262t_unit @ G2 @ H2 @ X ) @ ( partia966996272515721803t_unit @ G2 ) ) ) ) ) ).
% abelian_monoid.a_r_coset_subset_G
thf(fact_1108_abelian__monoid_Oa__r__coset__subset__G,axiom,
! [G2: partia4692342223508353374t_unit,H2: set_nat,X: nat] :
( ( abelia362511065248671243t_unit @ G2 )
=> ( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ord_less_eq_set_nat @ ( a_r_co9089071853028257912t_unit @ G2 @ H2 @ X ) @ ( partia3499330772048238685t_unit @ G2 ) ) ) ) ) ).
% abelian_monoid.a_r_coset_subset_G
thf(fact_1109_ring_Ospace__subgroup__props_I2_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ E ) ) ) ) ).
% ring.space_subgroup_props(2)
thf(fact_1110_ring_Ospace__subgroup__props_I2_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ E ) ) ) ) ).
% ring.space_subgroup_props(2)
thf(fact_1111_ring_Ospace__subgroup__props_I3_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int,V1: set_int,V22: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( ( member_set_int @ V1 @ E )
=> ( ( member_set_int @ V22 @ E )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R2 @ V1 @ V22 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(3)
thf(fact_1112_ring_Ospace__subgroup__props_I3_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat,V1: nat,V22: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( ( member_nat @ V1 @ E )
=> ( ( member_nat @ V22 @ E )
=> ( member_nat @ ( add_nat_Product_unit @ R2 @ V1 @ V22 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(3)
thf(fact_1113_ring_Ospace__subgroup__props_I5_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int,K: set_int,V2: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( ( member_set_int @ K @ K2 )
=> ( ( member_set_int @ V2 @ E )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ K @ V2 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(5)
thf(fact_1114_ring_Ospace__subgroup__props_I5_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat,K: nat,V2: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( ( member_nat @ K @ K2 )
=> ( ( member_nat @ V2 @ E )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ K @ V2 ) @ E ) ) ) ) ) ) ).
% ring.space_subgroup_props(5)
thf(fact_1115_ring_Otelescopic__base,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,F2: set_set_int,N: nat,M: nat,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( subfie3888952257595785920t_unit @ F2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ F2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ M @ F2 @ E )
=> ( embedd646006463343340164t_unit @ R2 @ ( times_times_nat @ N @ M ) @ K2 @ E ) ) ) ) ) ) ).
% ring.telescopic_base
thf(fact_1116_ring_Otelescopic__base,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,F2: set_nat,N: nat,M: nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( subfie4892355163478727762t_unit @ F2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ F2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ M @ F2 @ E )
=> ( embedd5688180257602933782t_unit @ R2 @ ( times_times_nat @ N @ M ) @ K2 @ E ) ) ) ) ) ) ).
% ring.telescopic_base
thf(fact_1117_ring_Ospace__subgroup__props_I4_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int,V2: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( ( member_set_int @ V2 @ E )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R2 @ V2 ) @ E ) ) ) ) ) ).
% ring.space_subgroup_props(4)
thf(fact_1118_ring_Ospace__subgroup__props_I4_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat,V2: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( ( member_nat @ V2 @ E )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ R2 @ V2 ) @ E ) ) ) ) ) ).
% ring.space_subgroup_props(4)
thf(fact_1119_ring_Ounique__dimension,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd8246663962306818995t_unit @ R2 @ K2 @ E )
=> ? [X4: nat] :
( ( embedd646006463343340164t_unit @ R2 @ X4 @ K2 @ E )
& ! [Y5: nat] :
( ( embedd646006463343340164t_unit @ R2 @ Y5 @ K2 @ E )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% ring.unique_dimension
thf(fact_1120_ring_Ounique__dimension,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd6096501799845681477t_unit @ R2 @ K2 @ E )
=> ? [X4: nat] :
( ( embedd5688180257602933782t_unit @ R2 @ X4 @ K2 @ E )
& ! [Y5: nat] :
( ( embedd5688180257602933782t_unit @ R2 @ Y5 @ K2 @ E )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% ring.unique_dimension
thf(fact_1121_abelian__group_Oa__coset__add__zero,axiom,
! [G2: partia4934656038542163276t_unit,M2: set_set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( a_r_co692709266861932262t_unit @ G2 @ M2 @ ( zero_s6269048424454532197t_unit @ G2 ) )
= M2 ) ) ) ).
% abelian_group.a_coset_add_zero
thf(fact_1122_abelian__group_Oa__coset__add__zero,axiom,
! [G2: partia4692342223508353374t_unit,M2: set_nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( a_r_co9089071853028257912t_unit @ G2 @ M2 @ ( zero_n5149899317435570679t_unit @ G2 ) )
= M2 ) ) ) ).
% abelian_group.a_coset_add_zero
thf(fact_1123_abelian__group_Oa__rcosI,axiom,
! [G2: partia4934656038542163276t_unit,H: set_int,H2: set_set_int,X: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( member_set_int @ H @ H2 )
=> ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ G2 @ H @ X ) @ ( a_r_co692709266861932262t_unit @ G2 @ H2 @ X ) ) ) ) ) ) ).
% abelian_group.a_rcosI
thf(fact_1124_abelian__group_Oa__rcosI,axiom,
! [G2: partia4692342223508353374t_unit,H: nat,H2: set_nat,X: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( member_nat @ H @ H2 )
=> ( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( member_nat @ ( add_nat_Product_unit @ G2 @ H @ X ) @ ( a_r_co9089071853028257912t_unit @ G2 @ H2 @ X ) ) ) ) ) ) ).
% abelian_group.a_rcosI
thf(fact_1125_abelian__group_Oa__coset__add__assoc,axiom,
! [G2: partia4934656038542163276t_unit,M2: set_set_int,G: set_int,H: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ G @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ H @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( a_r_co692709266861932262t_unit @ G2 @ ( a_r_co692709266861932262t_unit @ G2 @ M2 @ G ) @ H )
= ( a_r_co692709266861932262t_unit @ G2 @ M2 @ ( add_se5859248395121729892t_unit @ G2 @ G @ H ) ) ) ) ) ) ) ).
% abelian_group.a_coset_add_assoc
thf(fact_1126_abelian__group_Oa__coset__add__assoc,axiom,
! [G2: partia4692342223508353374t_unit,M2: set_nat,G: nat,H: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ G @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ H @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( a_r_co9089071853028257912t_unit @ G2 @ ( a_r_co9089071853028257912t_unit @ G2 @ M2 @ G ) @ H )
= ( a_r_co9089071853028257912t_unit @ G2 @ M2 @ ( add_nat_Product_unit @ G2 @ G @ H ) ) ) ) ) ) ) ).
% abelian_group.a_coset_add_assoc
thf(fact_1127_abelian__group_Oa__setmult__rcos__assoc,axiom,
! [G2: partia4934656038542163276t_unit,H2: set_set_int,K2: set_set_int,X: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( ord_le4403425263959731960et_int @ K2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( set_ad273131178244904872t_unit @ G2 @ H2 @ ( a_r_co692709266861932262t_unit @ G2 @ K2 @ X ) )
= ( a_r_co692709266861932262t_unit @ G2 @ ( set_ad273131178244904872t_unit @ G2 @ H2 @ K2 ) @ X ) ) ) ) ) ) ).
% abelian_group.a_setmult_rcos_assoc
thf(fact_1128_abelian__group_Oa__setmult__rcos__assoc,axiom,
! [G2: partia4692342223508353374t_unit,H2: set_nat,K2: set_nat,X: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( set_ad320919470248169786t_unit @ G2 @ H2 @ ( a_r_co9089071853028257912t_unit @ G2 @ K2 @ X ) )
= ( a_r_co9089071853028257912t_unit @ G2 @ ( set_ad320919470248169786t_unit @ G2 @ H2 @ K2 ) @ X ) ) ) ) ) ) ).
% abelian_group.a_setmult_rcos_assoc
thf(fact_1129_ring_Ospace__subgroup__props_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% ring.space_subgroup_props(1)
thf(fact_1130_ring_Ospace__subgroup__props_I1_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( ord_less_eq_set_nat @ E @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% ring.space_subgroup_props(1)
thf(fact_1131_ring_OSuc__dim,axiom,
! [R2: partia4934656038542163276t_unit,V2: set_int,E: set_set_int,N: nat,K2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( member_set_int @ V2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ~ ( member_set_int @ V2 @ E )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( embedd646006463343340164t_unit @ R2 @ ( suc @ N ) @ K2 @ ( embedd4283282269743769663t_unit @ R2 @ K2 @ V2 @ E ) ) ) ) ) ) ).
% ring.Suc_dim
thf(fact_1132_ring_OSuc__dim,axiom,
! [R2: partia4692342223508353374t_unit,V2: nat,E: set_nat,N: nat,K2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( member_nat @ V2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ~ ( member_nat @ V2 @ E )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( embedd5688180257602933782t_unit @ R2 @ ( suc @ N ) @ K2 @ ( embedd838748496991043025t_unit @ R2 @ K2 @ V2 @ E ) ) ) ) ) ) ).
% ring.Suc_dim
thf(fact_1133_abelian__group_Oa__coset__add__inv1,axiom,
! [G2: partia4934656038542163276t_unit,M2: set_set_int,X: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( ( a_r_co692709266861932262t_unit @ G2 @ M2 @ ( add_se5859248395121729892t_unit @ G2 @ X @ ( a_inv_5951419416477254493t_unit @ G2 @ Y ) ) )
= M2 )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( a_r_co692709266861932262t_unit @ G2 @ M2 @ X )
= ( a_r_co692709266861932262t_unit @ G2 @ M2 @ Y ) ) ) ) ) ) ) ).
% abelian_group.a_coset_add_inv1
thf(fact_1134_abelian__group_Oa__coset__add__inv1,axiom,
! [G2: partia4692342223508353374t_unit,M2: set_nat,X: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( ( a_r_co9089071853028257912t_unit @ G2 @ M2 @ ( add_nat_Product_unit @ G2 @ X @ ( a_inv_2472168910397739247t_unit @ G2 @ Y ) ) )
= M2 )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( a_r_co9089071853028257912t_unit @ G2 @ M2 @ X )
= ( a_r_co9089071853028257912t_unit @ G2 @ M2 @ Y ) ) ) ) ) ) ) ).
% abelian_group.a_coset_add_inv1
thf(fact_1135_abelian__group_Oa__coset__add__inv2,axiom,
! [G2: partia4934656038542163276t_unit,M2: set_set_int,X: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( ( a_r_co692709266861932262t_unit @ G2 @ M2 @ X )
= ( a_r_co692709266861932262t_unit @ G2 @ M2 @ Y ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( a_r_co692709266861932262t_unit @ G2 @ M2 @ ( add_se5859248395121729892t_unit @ G2 @ X @ ( a_inv_5951419416477254493t_unit @ G2 @ Y ) ) )
= M2 ) ) ) ) ) ) ).
% abelian_group.a_coset_add_inv2
thf(fact_1136_abelian__group_Oa__coset__add__inv2,axiom,
! [G2: partia4692342223508353374t_unit,M2: set_nat,X: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( ( a_r_co9089071853028257912t_unit @ G2 @ M2 @ X )
= ( a_r_co9089071853028257912t_unit @ G2 @ M2 @ Y ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( a_r_co9089071853028257912t_unit @ G2 @ M2 @ ( add_nat_Product_unit @ G2 @ X @ ( a_inv_2472168910397739247t_unit @ G2 @ Y ) ) )
= M2 ) ) ) ) ) ) ).
% abelian_group.a_coset_add_inv2
thf(fact_1137_abelian__group_Oa__rcos__assoc__lcos,axiom,
! [G2: partia4934656038542163276t_unit,H2: set_set_int,K2: set_set_int,X: set_int] :
( ( abelia23968383328945916t_unit @ G2 )
=> ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( ord_le4403425263959731960et_int @ K2 @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G2 ) )
=> ( ( set_ad273131178244904872t_unit @ G2 @ ( a_r_co692709266861932262t_unit @ G2 @ H2 @ X ) @ K2 )
= ( set_ad273131178244904872t_unit @ G2 @ H2 @ ( a_l_co3504123944629134560t_unit @ G2 @ X @ K2 ) ) ) ) ) ) ) ).
% abelian_group.a_rcos_assoc_lcos
thf(fact_1138_abelian__group_Oa__rcos__assoc__lcos,axiom,
! [G2: partia4692342223508353374t_unit,H2: set_nat,K2: set_nat,X: nat] :
( ( abelia406319425121669262t_unit @ G2 )
=> ( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( ord_less_eq_set_nat @ K2 @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G2 ) )
=> ( ( set_ad320919470248169786t_unit @ G2 @ ( a_r_co9089071853028257912t_unit @ G2 @ H2 @ X ) @ K2 )
= ( set_ad320919470248169786t_unit @ G2 @ H2 @ ( a_l_co3340896127261556338t_unit @ G2 @ X @ K2 ) ) ) ) ) ) ) ).
% abelian_group.a_rcos_assoc_lcos
thf(fact_1139_ring_Odimension__backwards,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ ( suc @ N ) @ K2 @ E )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
& ? [E3: set_set_int] :
( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E3 )
& ~ ( member_set_int @ X4 @ E3 )
& ( E
= ( embedd4283282269743769663t_unit @ R2 @ K2 @ X4 @ E3 ) ) ) ) ) ) ) ).
% ring.dimension_backwards
thf(fact_1140_ring_Odimension__backwards,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ ( suc @ N ) @ K2 @ E )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ R2 ) )
& ? [E3: set_nat] :
( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E3 )
& ~ ( member_nat @ X4 @ E3 )
& ( E
= ( embedd838748496991043025t_unit @ R2 @ K2 @ X4 @ E3 ) ) ) ) ) ) ) ).
% ring.dimension_backwards
thf(fact_1141_ring_Ospace__subgroup__props_I6_J,axiom,
! [R2: partia4934656038542163276t_unit,K2: set_set_int,N: nat,E: set_set_int,K: set_int,A: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( subfie3888952257595785920t_unit @ K2 @ R2 )
=> ( ( embedd646006463343340164t_unit @ R2 @ N @ K2 @ E )
=> ( ( member_set_int @ K @ ( minus_8897228262479074673et_int @ K2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ ( mult_s3864001451298473021t_unit @ R2 @ K @ A ) @ E )
=> ( member_set_int @ A @ E ) ) ) ) ) ) ) ).
% ring.space_subgroup_props(6)
thf(fact_1142_ring_Ospace__subgroup__props_I6_J,axiom,
! [R2: partia4692342223508353374t_unit,K2: set_nat,N: nat,E: set_nat,K: nat,A: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( subfie4892355163478727762t_unit @ K2 @ R2 )
=> ( ( embedd5688180257602933782t_unit @ R2 @ N @ K2 @ E )
=> ( ( member_nat @ K @ ( minus_minus_set_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R2 ) @ bot_bot_set_nat ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ ( mult_n6028127365542633569t_unit @ R2 @ K @ A ) @ E )
=> ( member_nat @ A @ E ) ) ) ) ) ) ) ).
% ring.space_subgroup_props(6)
thf(fact_1143_s_Odimension_Ocases,axiom,
! [A1: nat,A22: set_nat,A32: set_nat] :
( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) )
=> ~ ! [V3: nat,E4: set_nat,N3: nat] :
( ( A1
= ( suc @ N3 ) )
=> ( ( A32
= ( embedd838748496991043025t_unit @ ( mod_ring @ n ) @ A22 @ V3 @ E4 ) )
=> ( ( member_nat @ V3 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ~ ( member_nat @ V3 @ E4 )
=> ~ ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N3 @ A22 @ E4 ) ) ) ) ) ) ) ).
% s.dimension.cases
thf(fact_1144_s_Odimension_Osimps,axiom,
! [A1: nat,A22: set_nat,A32: set_nat] :
( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ A1 @ A22 @ A32 )
= ( ? [K5: set_nat] :
( ( A1 = zero_zero_nat )
& ( A22 = K5 )
& ( A32
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) )
| ? [V4: nat,E5: set_nat,N4: nat,K5: set_nat] :
( ( A1
= ( suc @ N4 ) )
& ( A22 = K5 )
& ( A32
= ( embedd838748496991043025t_unit @ ( mod_ring @ n ) @ K5 @ V4 @ E5 ) )
& ( member_nat @ V4 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
& ~ ( member_nat @ V4 @ E5 )
& ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N4 @ K5 @ E5 ) ) ) ) ).
% s.dimension.simps
thf(fact_1145_s_Ozero__dim,axiom,
! [K2: set_nat] : ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ zero_zero_nat @ K2 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) ).
% s.zero_dim
thf(fact_1146_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1147_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_1148_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1149_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_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_1157_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_1158_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_1159_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_1160_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_1161_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1162_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_1163_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_1164_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_1165_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_1166_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_1167_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_1168_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_1169_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1170_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1171_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1172_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1173_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1174_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1175_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1176_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1177_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1178_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_1179_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_1180_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_1181_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1182_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_1183_s_Odimension__zero,axiom,
! [K2: set_nat,E: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ zero_zero_nat @ K2 @ E )
=> ( E
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) ) ) ) ).
% s.dimension_zero
thf(fact_1184_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_1185_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_1186_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_1187_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_1188_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_1189_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_1190_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_1191_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_1192_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_1193_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1194_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_1195_s_Oadd_Onat__pow__0,axiom,
! [X: nat] :
( ( add_po2422570615063001561it_nat @ ( mod_ring @ n ) @ zero_zero_nat @ X )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ).
% s.add.nat_pow_0
thf(fact_1196_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1197_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_1198_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1199_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1200_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_1201_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_1202_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1203_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1204_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1205_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1206_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1207_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1208_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P @ X4 @ Y3 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1209_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1210_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1211_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1212_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1213_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1214_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1215_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_1216_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1217_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1218_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_1219_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_1220_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_1221_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1222_s_Oa__rcosetsI,axiom,
! [H2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ H2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) )
=> ( member_set_nat @ ( a_r_co9089071853028257912t_unit @ ( mod_ring @ n ) @ H2 @ X ) @ ( a_RCOS6328597822339572043t_unit @ ( mod_ring @ n ) @ H2 ) ) ) ) ).
% s.a_rcosetsI
thf(fact_1223_s_Odimension__direct__sum__space,axiom,
! [K2: set_nat,N: nat,E: set_nat,M: nat,F2: set_nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ M @ K2 @ F2 )
=> ( ( ( inf_inf_set_nat @ E @ F2 )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) @ bot_bot_set_nat ) )
=> ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ ( plus_plus_nat @ N @ M ) @ K2 @ ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ E @ F2 ) ) ) ) ) ) ).
% s.dimension_direct_sum_space
thf(fact_1224_s_Oint__embed__zero,axiom,
( ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ zero_zero_int )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ).
% s.int_embed_zero
thf(fact_1225_s_Osubring__inter,axiom,
! [I2: set_nat,J2: set_nat] :
( ( subrin2893992908230074586t_unit @ I2 @ ( mod_ring @ n ) )
=> ( ( subrin2893992908230074586t_unit @ J2 @ ( mod_ring @ n ) )
=> ( subrin2893992908230074586t_unit @ ( inf_inf_set_nat @ I2 @ J2 ) @ ( mod_ring @ n ) ) ) ) ).
% s.subring_inter
thf(fact_1226_s_Osubalgebra__inter,axiom,
! [K2: set_nat,V: set_nat,V5: set_nat] :
( ( embedd2609395410403458802t_unit @ K2 @ V @ ( mod_ring @ n ) )
=> ( ( embedd2609395410403458802t_unit @ K2 @ V5 @ ( mod_ring @ n ) )
=> ( embedd2609395410403458802t_unit @ K2 @ ( inf_inf_set_nat @ V @ V5 ) @ ( mod_ring @ n ) ) ) ) ).
% s.subalgebra_inter
thf(fact_1227_s_Osubcring__inter,axiom,
! [I2: set_nat,J2: set_nat] :
( ( subcri1627753237249443161t_unit @ I2 @ ( mod_ring @ n ) )
=> ( ( subcri1627753237249443161t_unit @ J2 @ ( mod_ring @ n ) )
=> ( subcri1627753237249443161t_unit @ ( inf_inf_set_nat @ I2 @ J2 ) @ ( mod_ring @ n ) ) ) ) ).
% s.subcring_inter
thf(fact_1228_s_Odimension__sum__space,axiom,
! [K2: set_nat,N: nat,E: set_nat,M: nat,F2: set_nat,K: nat] :
( ( subfie4892355163478727762t_unit @ K2 @ ( mod_ring @ n ) )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ N @ K2 @ E )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ M @ K2 @ F2 )
=> ( ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ K @ K2 @ ( inf_inf_set_nat @ E @ F2 ) )
=> ( embedd5688180257602933782t_unit @ ( mod_ring @ n ) @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) @ K2 @ ( set_ad320919470248169786t_unit @ ( mod_ring @ n ) @ E @ F2 ) ) ) ) ) ) ).
% s.dimension_sum_space
thf(fact_1229_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1230_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1231_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1232_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1233_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1234_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1235_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1236_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1237_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1238_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1239_s_Oembed__char__eq__0,axiom,
( ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ ( semiri1314217659103216013at_int @ ( ring_c2111903209400434062t_unit @ ( mod_ring @ n ) ) ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) ) ).
% s.embed_char_eq_0
thf(fact_1240_s_Ochar__bound_I1_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
=> ( ord_less_eq_nat @ ( ring_c2111903209400434062t_unit @ ( mod_ring @ n ) ) @ X ) ) ) ).
% s.char_bound(1)
thf(fact_1241_s_Ochar__bound_I2_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i4887507503946925882t_unit @ ( mod_ring @ n ) @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_n5149899317435570679t_unit @ ( mod_ring @ n ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c2111903209400434062t_unit @ ( mod_ring @ n ) ) ) ) ) ).
% s.char_bound(2)
thf(fact_1242_n__ge__0,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% n_ge_0
thf(fact_1243_n__ge__1,axiom,
ord_less_nat @ one_one_nat @ n ).
% n_ge_1
thf(fact_1244_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_1245_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1246_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1247_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1248_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1249_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_1250_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_1251_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1252_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1253_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_1254_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_1255_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_1256_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_1257_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_1258_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_1259_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_1260_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K4 )
=> ~ ( P @ I3 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1261_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_1262_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_1263_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1264_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1265_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1266_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1267_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1268_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1269_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1270_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1271_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1272_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1273_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1274_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_1275_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
% Conjectures (1)
thf(conj_0,conjecture,
member_nat_set_int @ ( ring_zfact_iso @ n ) @ ( ring_i1863809825068120638t_unit @ ( mod_ring @ n ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
%------------------------------------------------------------------------------