TPTP Problem File: SLH0649^1.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_00196_007035__18320198_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1456 ( 346 unt; 181 typ; 0 def)
% Number of atoms : 4111 (1161 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 16738 ( 281 ~; 30 |; 167 &;14047 @)
% ( 0 <=>;2213 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 15 ( 14 usr)
% Number of type conns : 669 ( 669 >; 0 *; 0 +; 0 <<)
% Number of symbols : 170 ( 167 usr; 12 con; 0-4 aty)
% Number of variables : 3222 ( 253 ^;2898 !; 71 ?;3222 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:42:06.984
%------------------------------------------------------------------------------
% Could-be-implicit typings (14)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_It__Int__Oint_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__Int__Oint_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J_J_J,type,
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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__Set__Oset_It__Int__Oint_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,
set_set_int_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
set_nat_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
set_set_set_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
list_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (167)
thf(sy_c_AbelCoset_OA__RCOSETS_001t__Nat__Onat_001t__Product____Type__Ounit,type,
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thf(sy_c_AbelCoset_Oadditive__subgroup_001t__Nat__Onat_001t__Product____Type__Ounit,type,
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thf(sy_c_AbelCoset_Oadditive__subgroup_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_AbelCoset_Oset__add_001t__Nat__Onat_001t__Product____Type__Ounit,type,
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thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Nat__Onat_001t__Group__Omonoid__Omonoid____ext_It__Nat__Onat_Mt__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
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thf(sy_c_Coset_Oorder_001t__Set__Oset_It__Int__Oint_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Divisibility_Omonoid__cancel_001t__Nat__Onat_001t__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Divisibility_Omonoid__cancel_001t__Set__Oset_It__Int__Oint_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Divisibility_Oproperfactor_001t__Set__Oset_It__Int__Oint_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Embedded__Algebras_Oring_Odimension_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Embedded__Algebras_Oring_Ofinite__dimension_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001t__Nat__Onat_001t__Product____Type__Ounit,type,
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thf(sy_c_Embedded__Algebras_Osubalgebra_001t__Nat__Onat_001t__Product____Type__Ounit,type,
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thf(sy_c_Field_Omod__ring,type,
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thf(sy_c_Field_Ozfact__iso__inv,type,
zfact_iso_inv: nat > set_int > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Int__Oint_J,type,
finite_card_set_int: set_set_int > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Group_OUnits_001t__Nat__Onat_001t__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Group_Omonoid_Omult_001t__Nat__Onat_001t__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
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mult_s3864001451298473021t_unit: partia4934656038542163276t_unit > set_int > set_int > set_int ).
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submon6016771835709735619t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_M_Eo_J,type,
minus_3277206198080951333_int_o: ( ( nat > set_int ) > $o ) > ( ( nat > set_int ) > $o ) > ( nat > set_int ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J_M_Eo_J,type,
minus_5750293070438678309_nat_o: ( ( set_int > nat ) > $o ) > ( ( set_int > nat ) > $o ) > ( set_int > nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Int__Oint_J_M_Eo_J,type,
minus_4665002624458620500_int_o: ( set_int > $o ) > ( set_int > $o ) > set_int > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J_J,type,
minus_5256904771846099296nt_nat: set_set_int_nat > set_set_int_nat > set_set_int_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
if_set_int: $o > set_int > set_int > set_int ).
thf(sy_c_IntRing_OZFact,type,
zFact: int > partia4934656038542163276t_unit ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_M_Eo_J,type,
ord_le2435888750224449770_int_o: ( ( nat > set_int ) > $o ) > ( ( nat > set_int ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_I_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le4908975622582176746_nat_o: ( ( set_int > nat ) > $o ) > ( ( set_int > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Int__Oint_J_M_Eo_J,type,
ord_less_set_int_o: ( set_int > $o ) > ( set_int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_M_Eo_J,type,
ord_le538297080920705635_int_o: ( set_set_int > $o ) > ( set_set_int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
ord_le2931775347370382171et_int: set_nat_set_int > set_nat_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J_J,type,
ord_le4941564535344212059nt_nat: set_set_int_nat > set_set_int_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
ord_less_set_set_int: set_set_int > set_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
ord_le4562804192517611682et_int: set_set_set_int > set_set_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_M_Eo_J,type,
ord_le8937012796915117022_int_o: ( ( nat > set_int ) > $o ) > ( ( nat > set_int ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le2186727632418068190_nat_o: ( ( set_int > nat ) > $o ) > ( ( set_int > nat ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Int__Oint_J_M_Eo_J,type,
ord_le1719207048323850125_int_o: ( set_int > $o ) > ( set_int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_M_Eo_J,type,
ord_le3688374355499189079_int_o: ( set_set_int > $o ) > ( set_set_int > $o ) > $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_I_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J_J,type,
ord_le8005464852987597927nt_nat: set_set_int_nat > set_set_int_nat > $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__Set__Oset_It__Int__Oint_J_J_J,type,
ord_le4317611570275147438et_int: set_set_set_int > set_set_set_int > $o ).
thf(sy_c_Polynomials_Oring_Opoly__of__const_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
poly_o1995067004181805532t_unit: partia4934656038542163276t_unit > set_int > list_set_int ).
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__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_Ocring_001t__Nat__Onat_001t__Product____Type__Ounit,type,
cring_4736503688146807908t_unit: partia4692342223508353374t_unit > $o ).
thf(sy_c_Ring_Ocring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
cring_3079150759069666002t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
domain6183376680155302761t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
field_5943785737635511755t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Ofinsum_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
finsum1765041520989711135et_int: partia4934656038542163276t_unit > ( ( nat > set_int ) > set_int ) > set_nat_set_int > set_int ).
thf(sy_c_Ring_Ofinsum_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J,type,
finsum2309937991119568927nt_nat: partia4934656038542163276t_unit > ( ( set_int > nat ) > set_int ) > set_set_int_nat > set_int ).
thf(sy_c_Ring_Ofinsum_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Nat__Onat,type,
finsum5143355122227517342it_nat: partia4934656038542163276t_unit > ( nat > set_int ) > set_nat > set_int ).
thf(sy_c_Ring_Ofinsum_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Set__Oset_It__Int__Oint_J,type,
finsum2830217253030306736et_int: partia4934656038542163276t_unit > ( set_int > set_int ) > set_set_int > set_int ).
thf(sy_c_Ring_Ofinsum_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
finsum1798061883449901670et_int: partia4934656038542163276t_unit > ( set_set_int > set_int ) > set_set_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_Ozero__update_001t__Nat__Onat_001t__Product____Type__Ounit,type,
zero_u3727541067695463886t_unit: ( nat > nat ) > partia4692342223508353374t_unit > partia4692342223508353374t_unit ).
thf(sy_c_Ring_Oring_Ozero__update_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
zero_u8982883731493082172t_unit: ( set_int > set_int ) > partia4934656038542163276t_unit > partia4934656038542163276t_unit ).
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__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_c6147214092195050492t_unit: partia4934656038542163276t_unit > nat ).
thf(sy_c_Ring__Characteristic_Oint__embed_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_i2743490682209504680t_unit: partia4934656038542163276t_unit > int > 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_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J,type,
collect_set_int_nat: ( ( set_int > nat ) > $o ) > set_set_int_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
collect_set_set_int: ( set_set_int > $o ) > set_set_set_int ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Int__Oint_J,type,
pow_set_int: set_set_int > set_set_set_int ).
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_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Nat__Onat_J,type,
insert_set_int_nat: ( set_int > nat ) > set_set_int_nat > set_set_int_nat ).
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__Set__Oset_It__Int__Oint_J_J,type,
insert_set_set_int: set_set_int > set_set_set_int > set_set_set_int ).
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__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__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
subrin7689096310803670856t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_UnivPoly_Obound_001t__Nat__Onat,type,
bound_nat: nat > nat > ( nat > nat ) > $o ).
thf(sy_c_UnivPoly_Obound_001t__Set__Oset_It__Int__Oint_J,type,
bound_set_int: set_int > nat > ( nat > set_int ) > $o ).
thf(sy_c_UnivPoly_Oup_001t__Nat__Onat_001t__Product____Type__Ounit,type,
up_nat_Product_unit: partia4692342223508353374t_unit > set_nat_nat ).
thf(sy_c_UnivPoly_Oup_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
up_set1168727741560211120t_unit: partia4934656038542163276t_unit > set_nat_set_int ).
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_c_member_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
member_set_set_int: set_set_int > set_set_set_int > $o ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1271)
thf(fact_0_calculation,axiom,
( cring_4736503688146807908t_unit
@ ( zero_u3727541067695463886t_unit
@ ^ [Uu: nat] : ( zfact_iso_inv @ n @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
@ ( mod_ring @ n ) ) ) ).
% calculation
thf(fact_1_local_Osemiring__axioms,axiom,
semiri8708897239777792527t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% local.semiring_axioms
thf(fact_2_abelian__monoid__axioms,axiom,
abelia3815030880812984441t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% abelian_monoid_axioms
thf(fact_3_subring__props_I2_J,axiom,
! [K: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ K ) ) ).
% subring_props(2)
thf(fact_4_zero__divides,axiom,
! [A: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ A )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% zero_divides
thf(fact_5_add_Ofinprod__one__eqI,axiom,
! [A2: set_set_int,F: set_int > set_int] :
( ! [X: set_int] :
( ( member_set_int @ X @ A2 )
=> ( ( F @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
=> ( ( finsum2830217253030306736et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% add.finprod_one_eqI
thf(fact_6_add_Ofinprod__one__eqI,axiom,
! [A2: set_nat_set_int,F: ( nat > set_int ) > set_int] :
( ! [X: nat > set_int] :
( ( member_nat_set_int @ X @ A2 )
=> ( ( F @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
=> ( ( finsum1765041520989711135et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% add.finprod_one_eqI
thf(fact_7_add_Ofinprod__one__eqI,axiom,
! [A2: set_nat,F: nat > set_int] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( F @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
=> ( ( finsum5143355122227517342it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% add.finprod_one_eqI
thf(fact_8_add_Ofinprod__one__eqI,axiom,
! [A2: set_set_int_nat,F: ( set_int > nat ) > set_int] :
( ! [X: set_int > nat] :
( ( member_set_int_nat @ X @ A2 )
=> ( ( F @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
=> ( ( finsum2309937991119568927nt_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% add.finprod_one_eqI
thf(fact_9_add_Ofinprod__one__eqI,axiom,
! [A2: set_set_set_int,F: set_set_int > set_int] :
( ! [X: set_set_int] :
( ( member_set_set_int @ X @ A2 )
=> ( ( F @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
=> ( ( finsum1798061883449901670et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% add.finprod_one_eqI
thf(fact_10_local_Oring__axioms,axiom,
ring_s5316885176909347197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% local.ring_axioms
thf(fact_11_is__abelian__group,axiom,
abelia23968383328945916t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% is_abelian_group
thf(fact_12_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_13_associated__sym,axiom,
! [A: set_int,B: set_int] :
( ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ A ) ) ).
% associated_sym
thf(fact_14_ring_Ofold__congs_I4_J,axiom,
! [R: partia4934656038542163276t_unit,R2: partia4934656038542163276t_unit,V: set_int,F: set_int > set_int,F2: set_int > set_int] :
( ( R = R2 )
=> ( ( ( zero_s6269048424454532197t_unit @ R2 )
= V )
=> ( ! [V2: set_int] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( zero_u8982883731493082172t_unit @ F @ R )
= ( zero_u8982883731493082172t_unit @ F2 @ R2 ) ) ) ) ) ).
% ring.fold_congs(4)
thf(fact_15_ring_Ofold__congs_I4_J,axiom,
! [R: partia4692342223508353374t_unit,R2: partia4692342223508353374t_unit,V: nat,F: nat > nat,F2: nat > nat] :
( ( R = R2 )
=> ( ( ( zero_n5149899317435570679t_unit @ R2 )
= V )
=> ( ! [V2: nat] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( zero_u3727541067695463886t_unit @ F @ R )
= ( zero_u3727541067695463886t_unit @ F2 @ R2 ) ) ) ) ) ).
% ring.fold_congs(4)
thf(fact_16_ring_Ounfold__congs_I4_J,axiom,
! [R: partia4934656038542163276t_unit,R2: partia4934656038542163276t_unit,V: set_int,F: set_int > set_int,F2: set_int > set_int] :
( ( R = R2 )
=> ( ( ( zero_s6269048424454532197t_unit @ R2 )
= V )
=> ( ! [V2: set_int] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( zero_u8982883731493082172t_unit @ F @ R )
= ( zero_u8982883731493082172t_unit @ F2 @ R2 ) ) ) ) ) ).
% ring.unfold_congs(4)
thf(fact_17_ring_Ounfold__congs_I4_J,axiom,
! [R: partia4692342223508353374t_unit,R2: partia4692342223508353374t_unit,V: nat,F: nat > nat,F2: nat > nat] :
( ( R = R2 )
=> ( ( ( zero_n5149899317435570679t_unit @ R2 )
= V )
=> ( ! [V2: nat] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( zero_u3727541067695463886t_unit @ F @ R )
= ( zero_u3727541067695463886t_unit @ F2 @ R2 ) ) ) ) ) ).
% ring.unfold_congs(4)
thf(fact_18_finsum__zero,axiom,
! [A2: set_set_set_int] :
( ( finsum1798061883449901670et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) )
@ ^ [I: set_set_int] : ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
@ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% finsum_zero
thf(fact_19_finsum__zero,axiom,
! [A2: set_set_int_nat] :
( ( finsum2309937991119568927nt_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) )
@ ^ [I: set_int > nat] : ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
@ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% finsum_zero
thf(fact_20_finsum__zero,axiom,
! [A2: set_nat] :
( ( finsum5143355122227517342it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) )
@ ^ [I: nat] : ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
@ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% finsum_zero
thf(fact_21_finsum__zero,axiom,
! [A2: set_nat_set_int] :
( ( finsum1765041520989711135et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) )
@ ^ [I: nat > set_int] : ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
@ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% finsum_zero
thf(fact_22_finsum__zero,axiom,
! [A2: set_set_int] :
( ( finsum2830217253030306736et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) )
@ ^ [I: set_int] : ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
@ A2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% finsum_zero
thf(fact_23_cring_Oaxioms_I1_J,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ring_n9194430563101542159t_unit @ R3 ) ) ).
% cring.axioms(1)
thf(fact_24_cring_Oaxioms_I1_J,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ring_s5316885176909347197t_unit @ R3 ) ) ).
% cring.axioms(1)
thf(fact_25_semiring_Oaxioms_I1_J,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( abelia3815030880812984441t_unit @ R3 ) ) ).
% semiring.axioms(1)
thf(fact_26_abelian__group_Oaxioms_I1_J,axiom,
! [G: partia4934656038542163276t_unit] :
( ( abelia23968383328945916t_unit @ G )
=> ( abelia3815030880812984441t_unit @ G ) ) ).
% abelian_group.axioms(1)
thf(fact_27_cring_Ois__cring,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( cring_4736503688146807908t_unit @ R3 ) ) ).
% cring.is_cring
thf(fact_28_cring_Ois__cring,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( cring_3079150759069666002t_unit @ R3 ) ) ).
% cring.is_cring
thf(fact_29_ring_Ois__abelian__group,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( abelia23968383328945916t_unit @ R3 ) ) ).
% ring.is_abelian_group
thf(fact_30_abelian__monoid_Ofinsum__zero,axiom,
! [G: partia4934656038542163276t_unit,A2: set_set_set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( finsum1798061883449901670et_int @ G
@ ^ [I: set_set_int] : ( zero_s6269048424454532197t_unit @ G )
@ A2 )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ).
% abelian_monoid.finsum_zero
thf(fact_31_abelian__monoid_Ofinsum__zero,axiom,
! [G: partia4934656038542163276t_unit,A2: set_set_int_nat] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( finsum2309937991119568927nt_nat @ G
@ ^ [I: set_int > nat] : ( zero_s6269048424454532197t_unit @ G )
@ A2 )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ).
% abelian_monoid.finsum_zero
thf(fact_32_abelian__monoid_Ofinsum__zero,axiom,
! [G: partia4934656038542163276t_unit,A2: set_nat] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( finsum5143355122227517342it_nat @ G
@ ^ [I: nat] : ( zero_s6269048424454532197t_unit @ G )
@ A2 )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ).
% abelian_monoid.finsum_zero
thf(fact_33_abelian__monoid_Ofinsum__zero,axiom,
! [G: partia4934656038542163276t_unit,A2: set_nat_set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( finsum1765041520989711135et_int @ G
@ ^ [I: nat > set_int] : ( zero_s6269048424454532197t_unit @ G )
@ A2 )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ).
% abelian_monoid.finsum_zero
thf(fact_34_abelian__monoid_Ofinsum__zero,axiom,
! [G: partia4934656038542163276t_unit,A2: set_set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( finsum2830217253030306736et_int @ G
@ ^ [I: set_int] : ( zero_s6269048424454532197t_unit @ G )
@ A2 )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ).
% abelian_monoid.finsum_zero
thf(fact_35_ring__iso__imp__img__cring,axiom,
! [H: set_int > set_int,S: partia4934656038542163276t_unit] :
( ( member5205197933313416826et_int @ H @ ( ring_i7857697894474155344t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S ) )
=> ( cring_3079150759069666002t_unit
@ ( zero_u8982883731493082172t_unit
@ ^ [Uu: set_int] : ( H @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
@ S ) ) ) ).
% ring_iso_imp_img_cring
thf(fact_36_ring__iso__imp__img__cring,axiom,
! [H: set_int > nat,S: partia4692342223508353374t_unit] :
( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S ) )
=> ( cring_4736503688146807908t_unit
@ ( zero_u3727541067695463886t_unit
@ ^ [Uu: nat] : ( H @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
@ S ) ) ) ).
% ring_iso_imp_img_cring
thf(fact_37_ring__iso__imp__img__ring,axiom,
! [H: set_int > set_int,S: partia4934656038542163276t_unit] :
( ( member5205197933313416826et_int @ H @ ( ring_i7857697894474155344t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S ) )
=> ( ring_s5316885176909347197t_unit
@ ( zero_u8982883731493082172t_unit
@ ^ [Uu: set_int] : ( H @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
@ S ) ) ) ).
% ring_iso_imp_img_ring
thf(fact_38_ring__iso__imp__img__ring,axiom,
! [H: set_int > nat,S: partia4692342223508353374t_unit] :
( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S ) )
=> ( ring_n9194430563101542159t_unit
@ ( zero_u3727541067695463886t_unit
@ ^ [Uu: nat] : ( H @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
@ S ) ) ) ).
% ring_iso_imp_img_ring
thf(fact_39_divides__cong__r,axiom,
! [X2: set_int,Y: set_int,Y2: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
=> ( ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Y2 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y2 ) ) ) ) ).
% divides_cong_r
thf(fact_40_divides__cong__l,axiom,
! [X2: set_int,X3: set_int,Y: set_int] :
( ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ X3 )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X3 @ Y )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) ) ) ) ).
% divides_cong_l
thf(fact_41_divides__zero,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% divides_zero
thf(fact_42_telescopic__base__dim_I1_J,axiom,
! [K: set_set_int,F3: set_set_int,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( subfie3888952257595785920t_unit @ F3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ F3 )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F3 @ E )
=> ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ E ) ) ) ) ) ).
% telescopic_base_dim(1)
thf(fact_43_subring__props_I4_J,axiom,
! [K: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( K != bot_bot_set_set_int ) ) ).
% subring_props(4)
thf(fact_44_divides__trans,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ C )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C ) ) ) ) ).
% divides_trans
thf(fact_45_associated__trans,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ C )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C ) ) ) ) ) ).
% associated_trans
thf(fact_46_assoc__subst,axiom,
! [A: set_int,B: set_int,F: set_int > set_int] :
( ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ! [A3: set_int,B2: set_int] :
( ( ( member_set_int @ A3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A3 @ B2 ) )
=> ( ( member_set_int @ ( F @ A3 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( member_set_int @ ( F @ B2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( F @ A3 ) @ ( F @ B2 ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% assoc_subst
thf(fact_47_properfactor__divides,axiom,
! [A: set_int,B: set_int] :
( ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ).
% properfactor_divides
thf(fact_48_subring__props_I7_J,axiom,
! [K: set_set_int,H1: set_int,H2: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ H1 @ K )
=> ( ( member_set_int @ H2 @ K )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H1 @ H2 ) @ K ) ) ) ) ).
% subring_props(7)
thf(fact_49_is__cring,axiom,
cring_3079150759069666002t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% is_cring
thf(fact_50_add_Ol__cancel,axiom,
! [C: set_int,A: set_int,B: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C @ A )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C @ B ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( A = B ) ) ) ) ) ).
% add.l_cancel
thf(fact_51_add_Or__cancel,axiom,
! [A: set_int,C: set_int,B: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ C ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( A = B ) ) ) ) ) ).
% add.r_cancel
thf(fact_52_a__assoc,axiom,
! [X2: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_53_a__comm,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 ) ) ) ) ).
% a_comm
thf(fact_54_a__lcomm,axiom,
! [X2: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_55_carrier__not__empty,axiom,
( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= bot_bot_set_set_int ) ).
% carrier_not_empty
thf(fact_56_ring__iso__restrict,axiom,
! [F: set_int > nat,S: partia4692342223508353374t_unit,G2: set_int > nat] :
( ( member_set_int_nat @ F @ ( ring_i6162119212153773794t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S ) )
=> ( ! [R4: set_int] :
( ( member_set_int @ R4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( F @ R4 )
= ( G2 @ R4 ) ) )
=> ( member_set_int_nat @ G2 @ ( ring_i6162119212153773794t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S ) ) ) ) ).
% ring_iso_restrict
thf(fact_57_add_Oinv__comm,axiom,
! [X2: set_int,Y: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% add.inv_comm
thf(fact_58_add_Ol__inv__ex,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ X2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% add.l_inv_ex
thf(fact_59_add_Oone__unique,axiom,
! [U: set_int] :
( ( member_set_int @ U @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ U @ X )
= X ) )
=> ( U
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% add.one_unique
thf(fact_60_add_Or__inv__ex,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% add.r_inv_ex
thf(fact_61_local_Ominus__unique,axiom,
! [Y: set_int,X2: set_int,Y2: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_62_properfactor__cong__l,axiom,
! [X3: set_int,X2: set_int,Y: set_int] :
( ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X3 @ X2 )
=> ( ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X3 @ Y ) ) ) ) ) ) ).
% properfactor_cong_l
thf(fact_63_mem__Collect__eq,axiom,
! [A: set_int,P: set_int > $o] :
( ( member_set_int @ A @ ( collect_set_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_64_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_65_mem__Collect__eq,axiom,
! [A: set_int > nat,P: ( set_int > nat ) > $o] :
( ( member_set_int_nat @ A @ ( collect_set_int_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_66_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
! [A: set_set_int,P: set_set_int > $o] :
( ( member_set_set_int @ A @ ( collect_set_set_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A2: set_set_int] :
( ( collect_set_int
@ ^ [X4: set_int] : ( member_set_int @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A2: set_nat_set_int] :
( ( collect_nat_set_int
@ ^ [X4: nat > set_int] : ( member_nat_set_int @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A2: set_set_int_nat] :
( ( collect_set_int_nat
@ ^ [X4: set_int > nat] : ( member_set_int_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A2: set_set_set_int] :
( ( collect_set_set_int
@ ^ [X4: set_set_int] : ( member_set_set_int @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_73_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_74_Collect__cong,axiom,
! [P: set_set_int > $o,Q: set_set_int > $o] :
( ! [X: set_set_int] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_set_set_int @ P )
= ( collect_set_set_int @ Q ) ) ) ).
% Collect_cong
thf(fact_75_properfactor__cong__r,axiom,
! [X2: set_int,Y: set_int,Y2: set_int] :
( ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
=> ( ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Y2 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y2 ) ) ) ) ) ) ).
% properfactor_cong_r
thf(fact_76_properfactor__trans1,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ C )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C ) ) ) ) ) ).
% properfactor_trans1
thf(fact_77_properfactor__trans2,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ C )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C ) ) ) ) ) ).
% properfactor_trans2
thf(fact_78_zero__closed,axiom,
member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% zero_closed
thf(fact_79_local_Oadd_Oright__cancel,axiom,
! [X2: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ X2 ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_80_a__closed,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% a_closed
thf(fact_81_associated__refl,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ A ) ) ).
% associated_refl
thf(fact_82_divides__refl,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ A ) ) ).
% divides_refl
thf(fact_83_add_Ol__cancel__one,axiom,
! [X2: set_int,A: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ A )
= X2 )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_84_add_Ol__cancel__one_H,axiom,
! [X2: set_int,A: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( X2
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ A ) )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_85_add_Or__cancel__one,axiom,
! [X2: set_int,A: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ X2 )
= X2 )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_86_add_Or__cancel__one_H,axiom,
! [X2: set_int,A: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( X2
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ X2 ) )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_87_l__zero,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X2 )
= X2 ) ) ).
% l_zero
thf(fact_88_r__zero,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= X2 ) ) ).
% r_zero
thf(fact_89_finsum__empty,axiom,
! [F: ( set_int > nat ) > set_int] :
( ( finsum2309937991119568927nt_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ bot_bo1204028561185317019nt_nat )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% finsum_empty
thf(fact_90_finsum__empty,axiom,
! [F: nat > set_int] :
( ( finsum5143355122227517342it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ bot_bot_set_nat )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% finsum_empty
thf(fact_91_finsum__empty,axiom,
! [F: ( nat > set_int ) > set_int] :
( ( finsum1765041520989711135et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ bot_bo8417611410066262939et_int )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% finsum_empty
thf(fact_92_finsum__empty,axiom,
! [F: set_int > set_int] :
( ( finsum2830217253030306736et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ bot_bot_set_set_int )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% finsum_empty
thf(fact_93_finsum__empty,axiom,
! [F: set_set_int > set_int] :
( ( finsum1798061883449901670et_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F @ bot_bo2384636101374064866et_int )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% finsum_empty
thf(fact_94_ring_Oring__simprules_I1_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_95_ring_Oring__simprules_I1_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_96_ring_Oring__simprules_I7_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( add_se5859248395121729892t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_97_ring_Oring__simprules_I7_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_nat_Product_unit @ R3 @ X2 @ ( add_nat_Product_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_98_ring_Oring__simprules_I10_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ R3 @ Y @ X2 ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_99_ring_Oring__simprules_I10_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X2 @ Y )
= ( add_nat_Product_unit @ R3 @ Y @ X2 ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_100_ring_Oring__simprules_I22_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( add_se5859248395121729892t_unit @ R3 @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ R3 @ Y @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_101_ring_Oring__simprules_I22_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X2 @ ( add_nat_Product_unit @ R3 @ Y @ Z ) )
= ( add_nat_Product_unit @ R3 @ Y @ ( add_nat_Product_unit @ R3 @ X2 @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_102_cring_Ocring__simprules_I1_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ) ).
% cring.cring_simprules(1)
thf(fact_103_cring_Ocring__simprules_I1_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ) ).
% cring.cring_simprules(1)
thf(fact_104_cring_Ocring__simprules_I7_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( add_se5859248395121729892t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(7)
thf(fact_105_cring_Ocring__simprules_I7_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_nat_Product_unit @ R3 @ X2 @ ( add_nat_Product_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(7)
thf(fact_106_cring_Ocring__simprules_I10_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ R3 @ Y @ X2 ) ) ) ) ) ).
% cring.cring_simprules(10)
thf(fact_107_cring_Ocring__simprules_I10_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X2 @ Y )
= ( add_nat_Product_unit @ R3 @ Y @ X2 ) ) ) ) ) ).
% cring.cring_simprules(10)
thf(fact_108_cring_Ocring__simprules_I23_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( add_se5859248395121729892t_unit @ R3 @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ R3 @ Y @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(23)
thf(fact_109_cring_Ocring__simprules_I23_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X2 @ ( add_nat_Product_unit @ R3 @ Y @ Z ) )
= ( add_nat_Product_unit @ R3 @ Y @ ( add_nat_Product_unit @ R3 @ X2 @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(23)
thf(fact_110_abelian__groupE_I1_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_111_abelian__groupE_I1_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_112_abelian__groupE_I3_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( abelia23968383328945916t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( add_se5859248395121729892t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_113_abelian__groupE_I3_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( abelia406319425121669262t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_nat_Product_unit @ R3 @ X2 @ ( add_nat_Product_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_114_abelian__groupE_I4_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ R3 @ Y @ X2 ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_115_abelian__groupE_I4_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X2 @ Y )
= ( add_nat_Product_unit @ R3 @ Y @ X2 ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_116_abelian__monoid_Oa__closed,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ G @ X2 @ Y ) @ ( partia966996272515721803t_unit @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_117_abelian__monoid_Oa__closed,axiom,
! [G: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( abelia362511065248671243t_unit @ G )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G ) )
=> ( member_nat @ ( add_nat_Product_unit @ G @ X2 @ Y ) @ ( partia3499330772048238685t_unit @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_118_abelian__monoid_Oa__lcomm,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ X2 @ ( add_se5859248395121729892t_unit @ G @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ G @ Y @ ( add_se5859248395121729892t_unit @ G @ X2 @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_119_abelian__monoid_Oa__lcomm,axiom,
! [G: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( abelia362511065248671243t_unit @ G )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( add_nat_Product_unit @ G @ X2 @ ( add_nat_Product_unit @ G @ Y @ Z ) )
= ( add_nat_Product_unit @ G @ Y @ ( add_nat_Product_unit @ G @ X2 @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_120_abelian__monoid_Oa__assoc,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ ( add_se5859248395121729892t_unit @ G @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ G @ X2 @ ( add_se5859248395121729892t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_121_abelian__monoid_Oa__assoc,axiom,
! [G: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( abelia362511065248671243t_unit @ G )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( add_nat_Product_unit @ G @ ( add_nat_Product_unit @ G @ X2 @ Y ) @ Z )
= ( add_nat_Product_unit @ G @ X2 @ ( add_nat_Product_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_122_abelian__monoid_Oa__comm,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ G @ Y @ X2 ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_123_abelian__monoid_Oa__comm,axiom,
! [G: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( abelia362511065248671243t_unit @ G )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( add_nat_Product_unit @ G @ X2 @ Y )
= ( add_nat_Product_unit @ G @ Y @ X2 ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_124_abelian__monoidE_I1_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia3815030880812984441t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_125_abelian__monoidE_I1_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( abelia362511065248671243t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_126_abelian__monoidE_I3_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( abelia3815030880812984441t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( add_se5859248395121729892t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_127_abelian__monoidE_I3_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( abelia362511065248671243t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_nat_Product_unit @ R3 @ X2 @ ( add_nat_Product_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_128_abelian__monoidE_I5_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia3815030880812984441t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ R3 @ Y @ X2 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_129_abelian__monoidE_I5_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( abelia362511065248671243t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X2 @ Y )
= ( add_nat_Product_unit @ R3 @ Y @ X2 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_130_semiring_Osemiring__simprules_I1_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_131_semiring_Osemiring__simprules_I1_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_132_semiring_Osemiring__simprules_I5_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( add_se5859248395121729892t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_133_semiring_Osemiring__simprules_I5_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_nat_Product_unit @ R3 @ X2 @ ( add_nat_Product_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_134_semiring_Osemiring__simprules_I7_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ R3 @ Y @ X2 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_135_semiring_Osemiring__simprules_I7_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X2 @ Y )
= ( add_nat_Product_unit @ R3 @ Y @ X2 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_136_semiring_Osemiring__simprules_I12_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( add_se5859248395121729892t_unit @ R3 @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ R3 @ Y @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_137_semiring_Osemiring__simprules_I12_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X2 @ ( add_nat_Product_unit @ R3 @ Y @ Z ) )
= ( add_nat_Product_unit @ R3 @ Y @ ( add_nat_Product_unit @ R3 @ X2 @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_138_ring_Oring__simprules_I8_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( zero_s6269048424454532197t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% ring.ring_simprules(8)
thf(fact_139_ring_Oring__simprules_I8_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( zero_n5149899317435570679t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% ring.ring_simprules(8)
thf(fact_140_ring_Oring__simprules_I15_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( zero_s6269048424454532197t_unit @ R3 ) )
= X2 ) ) ) ).
% ring.ring_simprules(15)
thf(fact_141_ring_Oring__simprules_I15_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X2 @ ( zero_n5149899317435570679t_unit @ R3 ) )
= X2 ) ) ) ).
% ring.ring_simprules(15)
thf(fact_142_cring_Ocring__simprules_I8_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( zero_s6269048424454532197t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% cring.cring_simprules(8)
thf(fact_143_cring_Ocring__simprules_I8_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( zero_n5149899317435570679t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% cring.cring_simprules(8)
thf(fact_144_cring_Ocring__simprules_I16_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( zero_s6269048424454532197t_unit @ R3 ) )
= X2 ) ) ) ).
% cring.cring_simprules(16)
thf(fact_145_cring_Ocring__simprules_I16_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X2 @ ( zero_n5149899317435570679t_unit @ R3 ) )
= X2 ) ) ) ).
% cring.cring_simprules(16)
thf(fact_146_abelian__groupI,axiom,
! [R3: partia4934656038542163276t_unit] :
( ! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ! [Y3: set_int] :
( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R3 @ X @ Y3 ) @ ( partia966996272515721803t_unit @ R3 ) ) ) )
=> ( ( member_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ! [Y3: set_int] :
( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R3 ) )
=> ! [Z2: set_int] :
( ( member_set_int @ Z2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( add_se5859248395121729892t_unit @ R3 @ X @ Y3 ) @ Z2 )
= ( add_se5859248395121729892t_unit @ R3 @ X @ ( add_se5859248395121729892t_unit @ R3 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ! [Y3: set_int] :
( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X @ Y3 )
= ( add_se5859248395121729892t_unit @ R3 @ Y3 @ X ) ) ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( zero_s6269048424454532197t_unit @ R3 ) @ X )
= X ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ? [Xa: set_int] :
( ( member_set_int @ Xa @ ( partia966996272515721803t_unit @ R3 ) )
& ( ( add_se5859248395121729892t_unit @ R3 @ Xa @ X )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) )
=> ( abelia23968383328945916t_unit @ R3 ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_147_abelian__groupI,axiom,
! [R3: partia4692342223508353374t_unit] :
( ! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( add_nat_Product_unit @ R3 @ X @ Y3 ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) )
=> ( ( member_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ! [Z2: nat] :
( ( member_nat @ Z2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( add_nat_Product_unit @ R3 @ X @ Y3 ) @ Z2 )
= ( add_nat_Product_unit @ R3 @ X @ ( add_nat_Product_unit @ R3 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X @ Y3 )
= ( add_nat_Product_unit @ R3 @ Y3 @ X ) ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( zero_n5149899317435570679t_unit @ R3 ) @ X )
= X ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ ( partia3499330772048238685t_unit @ R3 ) )
& ( ( add_nat_Product_unit @ R3 @ Xa @ X )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) )
=> ( abelia406319425121669262t_unit @ R3 ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_148_abelian__groupE_I5_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia23968383328945916t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( zero_s6269048424454532197t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% abelian_groupE(5)
thf(fact_149_abelian__groupE_I5_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( abelia406319425121669262t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( zero_n5149899317435570679t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% abelian_groupE(5)
thf(fact_150_abelian__groupE_I6_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia23968383328945916t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ? [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
& ( ( add_se5859248395121729892t_unit @ R3 @ X @ X2 )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_151_abelian__groupE_I6_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( abelia406319425121669262t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ? [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
& ( ( add_nat_Product_unit @ R3 @ X @ X2 )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_152_abelian__monoidI,axiom,
! [R3: partia4934656038542163276t_unit] :
( ! [X: set_int,Y3: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R3 @ X @ Y3 ) @ ( partia966996272515721803t_unit @ R3 ) ) ) )
=> ( ( member_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ! [X: set_int,Y3: set_int,Z2: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( add_se5859248395121729892t_unit @ R3 @ X @ Y3 ) @ Z2 )
= ( add_se5859248395121729892t_unit @ R3 @ X @ ( add_se5859248395121729892t_unit @ R3 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( zero_s6269048424454532197t_unit @ R3 ) @ X )
= X ) )
=> ( ! [X: set_int,Y3: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X @ Y3 )
= ( add_se5859248395121729892t_unit @ R3 @ Y3 @ X ) ) ) )
=> ( abelia3815030880812984441t_unit @ R3 ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_153_abelian__monoidI,axiom,
! [R3: partia4692342223508353374t_unit] :
( ! [X: nat,Y3: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( add_nat_Product_unit @ R3 @ X @ Y3 ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) )
=> ( ( member_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ! [X: nat,Y3: nat,Z2: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( add_nat_Product_unit @ R3 @ X @ Y3 ) @ Z2 )
= ( add_nat_Product_unit @ R3 @ X @ ( add_nat_Product_unit @ R3 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( zero_n5149899317435570679t_unit @ R3 ) @ X )
= X ) )
=> ( ! [X: nat,Y3: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X @ Y3 )
= ( add_nat_Product_unit @ R3 @ Y3 @ X ) ) ) )
=> ( abelia362511065248671243t_unit @ R3 ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_154_abelian__monoid_Ominus__unique,axiom,
! [G: partia4934656038542163276t_unit,Y: set_int,X2: set_int,Y2: set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( ( add_se5859248395121729892t_unit @ G @ Y @ X2 )
= ( zero_s6269048424454532197t_unit @ G ) )
=> ( ( ( add_se5859248395121729892t_unit @ G @ X2 @ Y2 )
= ( zero_s6269048424454532197t_unit @ G ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y2 @ ( partia966996272515721803t_unit @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_155_abelian__monoid_Ominus__unique,axiom,
! [G: partia4692342223508353374t_unit,Y: nat,X2: nat,Y2: nat] :
( ( abelia362511065248671243t_unit @ G )
=> ( ( ( add_nat_Product_unit @ G @ Y @ X2 )
= ( zero_n5149899317435570679t_unit @ G ) )
=> ( ( ( add_nat_Product_unit @ G @ X2 @ Y2 )
= ( zero_n5149899317435570679t_unit @ G ) )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Y2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_156_abelian__monoid_Or__zero,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ X2 @ ( zero_s6269048424454532197t_unit @ G ) )
= X2 ) ) ) ).
% abelian_monoid.r_zero
thf(fact_157_abelian__monoid_Or__zero,axiom,
! [G: partia4692342223508353374t_unit,X2: nat] :
( ( abelia362511065248671243t_unit @ G )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( add_nat_Product_unit @ G @ X2 @ ( zero_n5149899317435570679t_unit @ G ) )
= X2 ) ) ) ).
% abelian_monoid.r_zero
thf(fact_158_abelian__monoid_Ol__zero,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ ( zero_s6269048424454532197t_unit @ G ) @ X2 )
= X2 ) ) ) ).
% abelian_monoid.l_zero
thf(fact_159_abelian__monoid_Ol__zero,axiom,
! [G: partia4692342223508353374t_unit,X2: nat] :
( ( abelia362511065248671243t_unit @ G )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( add_nat_Product_unit @ G @ ( zero_n5149899317435570679t_unit @ G ) @ X2 )
= X2 ) ) ) ).
% abelian_monoid.l_zero
thf(fact_160_abelian__monoidE_I4_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia3815030880812984441t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( zero_s6269048424454532197t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% abelian_monoidE(4)
thf(fact_161_abelian__monoidE_I4_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( abelia362511065248671243t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( zero_n5149899317435570679t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% abelian_monoidE(4)
thf(fact_162_semiring_Osemiring__simprules_I6_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( zero_s6269048424454532197t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_163_semiring_Osemiring__simprules_I6_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( zero_n5149899317435570679t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_164_semiring_Osemiring__simprules_I11_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( zero_s6269048424454532197t_unit @ R3 ) )
= X2 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_165_semiring_Osemiring__simprules_I11_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X2 @ ( zero_n5149899317435570679t_unit @ R3 ) )
= X2 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_166_ring_Oring__simprules_I2_J,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ).
% ring.ring_simprules(2)
thf(fact_167_ring_Oring__simprules_I2_J,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ).
% ring.ring_simprules(2)
thf(fact_168_cring_Ocring__simprules_I2_J,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ).
% cring.cring_simprules(2)
thf(fact_169_cring_Ocring__simprules_I2_J,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ).
% cring.cring_simprules(2)
thf(fact_170_abelian__groupE_I2_J,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( abelia23968383328945916t_unit @ R3 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ).
% abelian_groupE(2)
thf(fact_171_abelian__groupE_I2_J,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( abelia406319425121669262t_unit @ R3 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ).
% abelian_groupE(2)
thf(fact_172_abelian__monoid_Ozero__closed,axiom,
! [G: partia4934656038542163276t_unit] :
( ( abelia3815030880812984441t_unit @ G )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ G ) @ ( partia966996272515721803t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_173_abelian__monoid_Ozero__closed,axiom,
! [G: partia4692342223508353374t_unit] :
( ( abelia362511065248671243t_unit @ G )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ G ) @ ( partia3499330772048238685t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_174_abelian__monoidE_I2_J,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( abelia3815030880812984441t_unit @ R3 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ).
% abelian_monoidE(2)
thf(fact_175_abelian__monoidE_I2_J,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( abelia362511065248671243t_unit @ R3 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ).
% abelian_monoidE(2)
thf(fact_176_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_177_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( member_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_178_abelian__monoid_Ofinsum__empty,axiom,
! [G: partia4934656038542163276t_unit,F: ( set_int > nat ) > set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( finsum2309937991119568927nt_nat @ G @ F @ bot_bo1204028561185317019nt_nat )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_179_abelian__monoid_Ofinsum__empty,axiom,
! [G: partia4934656038542163276t_unit,F: nat > set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( finsum5143355122227517342it_nat @ G @ F @ bot_bot_set_nat )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_180_abelian__monoid_Ofinsum__empty,axiom,
! [G: partia4934656038542163276t_unit,F: ( nat > set_int ) > set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( finsum1765041520989711135et_int @ G @ F @ bot_bo8417611410066262939et_int )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_181_abelian__monoid_Ofinsum__empty,axiom,
! [G: partia4934656038542163276t_unit,F: set_int > set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( finsum2830217253030306736et_int @ G @ F @ bot_bot_set_set_int )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_182_abelian__monoid_Ofinsum__empty,axiom,
! [G: partia4934656038542163276t_unit,F: set_set_int > set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( finsum1798061883449901670et_int @ G @ F @ bot_bo2384636101374064866et_int )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_183_onepideal,axiom,
princi8860937869964495385t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% onepideal
thf(fact_184_isgcd__divides__l,axiom,
! [A: set_int,B: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( isgcd_4636411027072998995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ A @ B ) ) ) ) ).
% isgcd_divides_l
thf(fact_185_isgcd__divides__r,axiom,
! [B: set_int,A: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ A )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( isgcd_4636411027072998995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ A @ B ) ) ) ) ).
% isgcd_divides_r
thf(fact_186_finite__dimension__imp__subalgebra,axiom,
! [K: set_set_int,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ E )
=> ( embedd2743979684206749024t_unit @ K @ E @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% finite_dimension_imp_subalgebra
thf(fact_187_associated__iff__same__ideal,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
= ( ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A )
= ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B ) ) ) ) ) ).
% associated_iff_same_ideal
thf(fact_188_up__add__closed,axiom,
! [P2: nat > set_int,Q2: nat > set_int] :
( ( member_nat_set_int @ P2 @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_nat_set_int @ Q2 @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_nat_set_int
@ ^ [I: nat] : ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( P2 @ I ) @ ( Q2 @ I ) )
@ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% up_add_closed
thf(fact_189_cring_Oring__iso__imp__img__cring,axiom,
! [R3: partia4692342223508353374t_unit,H: nat > set_int,S: partia4934656038542163276t_unit] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R3 @ S ) )
=> ( cring_3079150759069666002t_unit
@ ( zero_u8982883731493082172t_unit
@ ^ [Uu: set_int] : ( H @ ( zero_n5149899317435570679t_unit @ R3 ) )
@ S ) ) ) ) ).
% cring.ring_iso_imp_img_cring
thf(fact_190_cring_Oring__iso__imp__img__cring,axiom,
! [R3: partia4934656038542163276t_unit,H: set_int > set_int,S: partia4934656038542163276t_unit] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member5205197933313416826et_int @ H @ ( ring_i7857697894474155344t_unit @ R3 @ S ) )
=> ( cring_3079150759069666002t_unit
@ ( zero_u8982883731493082172t_unit
@ ^ [Uu: set_int] : ( H @ ( zero_s6269048424454532197t_unit @ R3 ) )
@ S ) ) ) ) ).
% cring.ring_iso_imp_img_cring
thf(fact_191_cring_Oring__iso__imp__img__cring,axiom,
! [R3: partia4692342223508353374t_unit,H: nat > nat,S: partia4692342223508353374t_unit] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat_nat @ H @ ( ring_i3628375019190340560t_unit @ R3 @ S ) )
=> ( cring_4736503688146807908t_unit
@ ( zero_u3727541067695463886t_unit
@ ^ [Uu: nat] : ( H @ ( zero_n5149899317435570679t_unit @ R3 ) )
@ S ) ) ) ) ).
% cring.ring_iso_imp_img_cring
thf(fact_192_cring_Oring__iso__imp__img__cring,axiom,
! [R3: partia4934656038542163276t_unit,H: set_int > nat,S: partia4692342223508353374t_unit] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ R3 @ S ) )
=> ( cring_4736503688146807908t_unit
@ ( zero_u3727541067695463886t_unit
@ ^ [Uu: nat] : ( H @ ( zero_s6269048424454532197t_unit @ R3 ) )
@ S ) ) ) ) ).
% cring.ring_iso_imp_img_cring
thf(fact_193_ring_Oring__iso__imp__img__ring,axiom,
! [R3: partia4934656038542163276t_unit,H: set_int > set_int,S: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member5205197933313416826et_int @ H @ ( ring_i7857697894474155344t_unit @ R3 @ S ) )
=> ( ring_s5316885176909347197t_unit
@ ( zero_u8982883731493082172t_unit
@ ^ [Uu: set_int] : ( H @ ( zero_s6269048424454532197t_unit @ R3 ) )
@ S ) ) ) ) ).
% ring.ring_iso_imp_img_ring
thf(fact_194_ring_Oring__iso__imp__img__ring,axiom,
! [R3: partia4934656038542163276t_unit,H: set_int > nat,S: partia4692342223508353374t_unit] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ R3 @ S ) )
=> ( ring_n9194430563101542159t_unit
@ ( zero_u3727541067695463886t_unit
@ ^ [Uu: nat] : ( H @ ( zero_s6269048424454532197t_unit @ R3 ) )
@ S ) ) ) ) ).
% ring.ring_iso_imp_img_ring
thf(fact_195_subring__props_I1_J,axiom,
! [K: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% subring_props(1)
thf(fact_196_carrier__is__subcring,axiom,
subcri1024317279029940167t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% carrier_is_subcring
thf(fact_197_sum__space__dim_I1_J,axiom,
! [K: set_set_int,E: set_set_int,F3: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ E )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ F3 )
=> ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ E @ F3 ) ) ) ) ) ).
% sum_space_dim(1)
thf(fact_198_cgenideal__self,axiom,
! [I2: set_int] :
( ( member_set_int @ I2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ I2 @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 ) ) ) ).
% cgenideal_self
thf(fact_199_setadd__subset__G,axiom,
! [H3: set_set_int,K: set_set_int] :
( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H3 @ K ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% setadd_subset_G
thf(fact_200_set__add__comm,axiom,
! [I3: set_set_int,J: set_set_int] :
( ( ord_le4403425263959731960et_int @ I3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ J @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I3 @ J )
= ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ J @ I3 ) ) ) ) ).
% set_add_comm
thf(fact_201_set__add__closed,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ B3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A2 @ B3 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% set_add_closed
thf(fact_202_subalgebra__in__carrier,axiom,
! [K: set_set_int,V3: set_set_int] :
( ( embedd2743979684206749024t_unit @ K @ V3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ord_le4403425263959731960et_int @ V3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% subalgebra_in_carrier
thf(fact_203_carrier__is__subalgebra,axiom,
! [K: set_set_int] :
( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( embedd2743979684206749024t_unit @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% carrier_is_subalgebra
thf(fact_204_cgenideal__is__principalideal,axiom,
! [I2: set_int] :
( ( member_set_int @ I2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( princi8860937869964495385t_unit @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% cgenideal_is_principalideal
thf(fact_205_to__contain__is__to__divide,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A ) )
= ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ) ) ).
% to_contain_is_to_divide
thf(fact_206_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_207_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_208_subalbegra__incl__imp__finite__dimension,axiom,
! [K: set_set_int,E: set_set_int,V3: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ E )
=> ( ( embedd2743979684206749024t_unit @ K @ V3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ord_le4403425263959731960et_int @ V3 @ E )
=> ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ V3 ) ) ) ) ) ).
% subalbegra_incl_imp_finite_dimension
thf(fact_209_of__nat__mono,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).
% of_nat_mono
thf(fact_210_of__nat__mono,axiom,
! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).
% of_nat_mono
thf(fact_211_ring__iso__memE_I1_J,axiom,
! [H: set_int > set_int,R3: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,X2: set_int] :
( ( member5205197933313416826et_int @ H @ ( ring_i7857697894474155344t_unit @ R3 @ S ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( H @ X2 ) @ ( partia966996272515721803t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_212_ring__iso__memE_I1_J,axiom,
! [H: set_int > nat,R3: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,X2: set_int] :
( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ R3 @ S ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_nat @ ( H @ X2 ) @ ( partia3499330772048238685t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_213_ring__iso__memE_I1_J,axiom,
! [H: nat > set_int,R3: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,X2: nat] :
( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R3 @ S ) )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_set_int @ ( H @ X2 ) @ ( partia966996272515721803t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_214_ring__iso__memE_I1_J,axiom,
! [H: nat > nat,R3: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit,X2: nat] :
( ( member_nat_nat @ H @ ( ring_i3628375019190340560t_unit @ R3 @ S ) )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( H @ X2 ) @ ( partia3499330772048238685t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_215_ring__iso__memE_I3_J,axiom,
! [H: set_int > set_int,R3: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( member5205197933313416826et_int @ H @ ( ring_i7857697894474155344t_unit @ R3 @ S ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( H @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_216_ring__iso__memE_I3_J,axiom,
! [H: set_int > nat,R3: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,X2: set_int,Y: set_int] :
( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ R3 @ S ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( H @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) )
= ( add_nat_Product_unit @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_217_ring__iso__memE_I3_J,axiom,
! [H: nat > set_int,R3: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,X2: nat,Y: nat] :
( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R3 @ S ) )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( H @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_218_ring_Oring__iso__restrict,axiom,
! [R3: partia4934656038542163276t_unit,F: set_int > nat,S: partia4692342223508353374t_unit,G2: set_int > nat] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int_nat @ F @ ( ring_i6162119212153773794t_unit @ R3 @ S ) )
=> ( ! [R4: set_int] :
( ( member_set_int @ R4 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( F @ R4 )
= ( G2 @ R4 ) ) )
=> ( member_set_int_nat @ G2 @ ( ring_i6162119212153773794t_unit @ R3 @ S ) ) ) ) ) ).
% ring.ring_iso_restrict
thf(fact_219_bound__upD,axiom,
! [F: nat > set_int] :
( ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [N2: nat] : ( bound_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ N2 @ F ) ) ).
% bound_upD
thf(fact_220_up__minus__closed,axiom,
! [P2: nat > set_int,Q2: nat > set_int] :
( ( member_nat_set_int @ P2 @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_nat_set_int @ Q2 @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_nat_set_int
@ ^ [I: nat] : ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( P2 @ I ) @ ( Q2 @ I ) )
@ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% up_minus_closed
thf(fact_221_a__lcos__m__assoc,axiom,
! [M2: set_set_int,G2: set_int,H: set_int] :
( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ G2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ G2 @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H @ M2 ) )
= ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ G2 @ H ) @ M2 ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_222_a__lcos__mult__one,axiom,
! [M2: set_set_int] :
( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ M2 )
= M2 ) ) ).
% a_lcos_mult_one
thf(fact_223_add__additive__subgroups,axiom,
! [H3: set_set_int,K: set_set_int] :
( ( additi7073586575563672860t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( additi7073586575563672860t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( additi7073586575563672860t_unit @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H3 @ K ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% add_additive_subgroups
thf(fact_224_set__add__zero,axiom,
! [A2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ A2 )
= A2 ) ) ).
% set_add_zero
thf(fact_225_a__l__coset__subset__G,axiom,
! [H3: set_set_int,X2: set_int] :
( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ H3 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_226_subset__Idl__subset,axiom,
! [I3: set_set_int,H3: set_set_int] :
( ( ord_le4403425263959731960et_int @ I3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ H3 @ I3 )
=> ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H3 ) @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I3 ) ) ) ) ).
% subset_Idl_subset
thf(fact_227_genideal__self,axiom,
! [S: set_set_int] :
( ( ord_le4403425263959731960et_int @ S @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ S @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S ) ) ) ).
% genideal_self
thf(fact_228_genideal__self_H,axiom,
! [I2: set_int] :
( ( member_set_int @ I2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ I2 @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ I2 @ bot_bot_set_set_int ) ) ) ) ).
% genideal_self'
thf(fact_229_genideal__zero,axiom,
( ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) ).
% genideal_zero
thf(fact_230_zeropideal,axiom,
princi8860937869964495385t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% zeropideal
thf(fact_231_Idl__subset__ideal_H,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) )
= ( member_set_int @ A @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_232_cgenideal__eq__genideal,axiom,
! [I2: set_int] :
( ( member_set_int @ I2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 )
= ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ I2 @ bot_bot_set_set_int ) ) ) ) ).
% cgenideal_eq_genideal
thf(fact_233_minus__closed,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% minus_closed
thf(fact_234_r__right__minus__eq,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( A = B ) ) ) ) ).
% r_right_minus_eq
thf(fact_235_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_236_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_237_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_238_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_239_le__trans,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K2 )
=> ( ord_less_eq_nat @ I2 @ K2 ) ) ) ).
% le_trans
thf(fact_240_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_241_ring_Oring__simprules_I4_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( a_minu5974516859897376926t_unit @ R3 @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_242_ring_Oring__simprules_I4_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( a_minu1751788497103602224t_unit @ R3 @ X2 @ Y ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_243_cring_Ocring__simprules_I4_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( a_minu5974516859897376926t_unit @ R3 @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ) ).
% cring.cring_simprules(4)
thf(fact_244_cring_Ocring__simprules_I4_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( a_minu1751788497103602224t_unit @ R3 @ X2 @ Y ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ) ).
% cring.cring_simprules(4)
thf(fact_245_abelian__group_Ominus__closed,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( member_set_int @ ( a_minu5974516859897376926t_unit @ G @ X2 @ Y ) @ ( partia966996272515721803t_unit @ G ) ) ) ) ) ).
% abelian_group.minus_closed
thf(fact_246_abelian__group_Ominus__closed,axiom,
! [G: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G ) )
=> ( member_nat @ ( a_minu1751788497103602224t_unit @ G @ X2 @ Y ) @ ( partia3499330772048238685t_unit @ G ) ) ) ) ) ).
% abelian_group.minus_closed
thf(fact_247_mem__upI,axiom,
! [F: nat > set_int,R3: partia4934656038542163276t_unit] :
( ! [N2: nat] : ( member_set_int @ ( F @ N2 ) @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ? [N3: nat] : ( bound_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ N3 @ F )
=> ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ R3 ) ) ) ) ).
% mem_upI
thf(fact_248_mem__upI,axiom,
! [F: nat > nat,R3: partia4692342223508353374t_unit] :
( ! [N2: nat] : ( member_nat @ ( F @ N2 ) @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ? [N3: nat] : ( bound_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ N3 @ F )
=> ( member_nat_nat @ F @ ( up_nat_Product_unit @ R3 ) ) ) ) ).
% mem_upI
thf(fact_249_singleton__insert__inj__eq,axiom,
! [B: set_int,A: set_int,A2: set_set_int] :
( ( ( insert_set_int @ B @ bot_bot_set_set_int )
= ( insert_set_int @ A @ A2 ) )
= ( ( A = B )
& ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_250_singleton__insert__inj__eq,axiom,
! [B: set_set_int,A: set_set_int,A2: set_set_set_int] :
( ( ( insert_set_set_int @ B @ bot_bo2384636101374064866et_int )
= ( insert_set_set_int @ A @ A2 ) )
= ( ( A = B )
& ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ B @ bot_bo2384636101374064866et_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_251_singleton__insert__inj__eq_H,axiom,
! [A: set_int,A2: set_set_int,B: set_int] :
( ( ( insert_set_int @ A @ A2 )
= ( insert_set_int @ B @ bot_bot_set_set_int ) )
= ( ( A = B )
& ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_252_singleton__insert__inj__eq_H,axiom,
! [A: set_set_int,A2: set_set_set_int,B: set_set_int] :
( ( ( insert_set_set_int @ A @ A2 )
= ( insert_set_set_int @ B @ bot_bo2384636101374064866et_int ) )
= ( ( A = B )
& ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ B @ bot_bo2384636101374064866et_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_253_ring_Oset__add__zero,axiom,
! [R3: partia4934656038542163276t_unit,A2: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( ord_le4403425263959731960et_int @ A2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( set_ad273131178244904872t_unit @ R3 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ bot_bot_set_set_int ) @ A2 )
= A2 ) ) ) ).
% ring.set_add_zero
thf(fact_254_ring_Oset__add__zero,axiom,
! [R3: partia4692342223508353374t_unit,A2: set_nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( set_ad320919470248169786t_unit @ R3 @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ bot_bot_set_nat ) @ A2 )
= A2 ) ) ) ).
% ring.set_add_zero
thf(fact_255_cring_Ocgenideal__eq__genideal,axiom,
! [R3: partia4934656038542163276t_unit,I2: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ I2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( cgenid8502489213727343375t_unit @ R3 @ I2 )
= ( genide1545711809618862555t_unit @ R3 @ ( insert_set_int @ I2 @ bot_bot_set_set_int ) ) ) ) ) ).
% cring.cgenideal_eq_genideal
thf(fact_256_cring_Ocgenideal__eq__genideal,axiom,
! [R3: partia4692342223508353374t_unit,I2: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ I2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( cgenid8693976350862946099t_unit @ R3 @ I2 )
= ( genide4496968333291595629t_unit @ R3 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) ) ) ) ).
% cring.cgenideal_eq_genideal
thf(fact_257_ring_OIdl__subset__ideal_H,axiom,
! [R3: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ R3 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) @ ( genide1545711809618862555t_unit @ R3 @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) )
= ( member_set_int @ A @ ( genide1545711809618862555t_unit @ R3 @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_258_ring_OIdl__subset__ideal_H,axiom,
! [R3: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( ord_less_eq_set_nat @ ( genide4496968333291595629t_unit @ R3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ ( genide4496968333291595629t_unit @ R3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
= ( member_nat @ A @ ( genide4496968333291595629t_unit @ R3 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_259_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_260_singleton__conv2,axiom,
! [A: set_int] :
( ( collect_set_int
@ ( ^ [Y5: set_int,Z3: set_int] : ( Y5 = Z3 )
@ A ) )
= ( insert_set_int @ A @ bot_bot_set_set_int ) ) ).
% singleton_conv2
thf(fact_261_singleton__conv2,axiom,
! [A: set_set_int] :
( ( collect_set_set_int
@ ( ^ [Y5: set_set_int,Z3: set_set_int] : ( Y5 = Z3 )
@ A ) )
= ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) ).
% singleton_conv2
thf(fact_262_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X4: nat] : ( X4 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_263_singleton__conv,axiom,
! [A: set_int] :
( ( collect_set_int
@ ^ [X4: set_int] : ( X4 = A ) )
= ( insert_set_int @ A @ bot_bot_set_set_int ) ) ).
% singleton_conv
thf(fact_264_singleton__conv,axiom,
! [A: set_set_int] :
( ( collect_set_set_int
@ ^ [X4: set_set_int] : ( X4 = A ) )
= ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) ).
% singleton_conv
thf(fact_265_boundD__carrier,axiom,
! [N: nat,F: nat > set_int,M: nat] :
( ( bound_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_set_int @ ( F @ M ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% boundD_carrier
thf(fact_266_subsetI,axiom,
! [A2: set_nat_set_int,B3: set_nat_set_int] :
( ! [X: nat > set_int] :
( ( member_nat_set_int @ X @ A2 )
=> ( member_nat_set_int @ X @ B3 ) )
=> ( ord_le5995675665013768039et_int @ A2 @ B3 ) ) ).
% subsetI
thf(fact_267_subsetI,axiom,
! [A2: set_nat,B3: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B3 ) )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_268_subsetI,axiom,
! [A2: set_set_int_nat,B3: set_set_int_nat] :
( ! [X: set_int > nat] :
( ( member_set_int_nat @ X @ A2 )
=> ( member_set_int_nat @ X @ B3 ) )
=> ( ord_le8005464852987597927nt_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_269_subsetI,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ! [X: set_int] :
( ( member_set_int @ X @ A2 )
=> ( member_set_int @ X @ B3 ) )
=> ( ord_le4403425263959731960et_int @ A2 @ B3 ) ) ).
% subsetI
thf(fact_270_subsetI,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ! [X: set_set_int] :
( ( member_set_set_int @ X @ A2 )
=> ( member_set_set_int @ X @ B3 ) )
=> ( ord_le4317611570275147438et_int @ A2 @ B3 ) ) ).
% subsetI
thf(fact_271_subset__antisym,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ( ( ord_le4403425263959731960et_int @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_272_subset__antisym,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ( ord_le4317611570275147438et_int @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_273_empty__iff,axiom,
! [C: nat > set_int] :
~ ( member_nat_set_int @ C @ bot_bo8417611410066262939et_int ) ).
% empty_iff
thf(fact_274_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_275_empty__iff,axiom,
! [C: set_int > nat] :
~ ( member_set_int_nat @ C @ bot_bo1204028561185317019nt_nat ) ).
% empty_iff
thf(fact_276_empty__iff,axiom,
! [C: set_int] :
~ ( member_set_int @ C @ bot_bot_set_set_int ) ).
% empty_iff
thf(fact_277_empty__iff,axiom,
! [C: set_set_int] :
~ ( member_set_set_int @ C @ bot_bo2384636101374064866et_int ) ).
% empty_iff
thf(fact_278_all__not__in__conv,axiom,
! [A2: set_nat_set_int] :
( ( ! [X4: nat > set_int] :
~ ( member_nat_set_int @ X4 @ A2 ) )
= ( A2 = bot_bo8417611410066262939et_int ) ) ).
% all_not_in_conv
thf(fact_279_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat @ X4 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_280_all__not__in__conv,axiom,
! [A2: set_set_int_nat] :
( ( ! [X4: set_int > nat] :
~ ( member_set_int_nat @ X4 @ A2 ) )
= ( A2 = bot_bo1204028561185317019nt_nat ) ) ).
% all_not_in_conv
thf(fact_281_all__not__in__conv,axiom,
! [A2: set_set_int] :
( ( ! [X4: set_int] :
~ ( member_set_int @ X4 @ A2 ) )
= ( A2 = bot_bot_set_set_int ) ) ).
% all_not_in_conv
thf(fact_282_all__not__in__conv,axiom,
! [A2: set_set_set_int] :
( ( ! [X4: set_set_int] :
~ ( member_set_set_int @ X4 @ A2 ) )
= ( A2 = bot_bo2384636101374064866et_int ) ) ).
% all_not_in_conv
thf(fact_283_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_284_Collect__empty__eq,axiom,
! [P: set_int > $o] :
( ( ( collect_set_int @ P )
= bot_bot_set_set_int )
= ( ! [X4: set_int] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_285_Collect__empty__eq,axiom,
! [P: set_set_int > $o] :
( ( ( collect_set_set_int @ P )
= bot_bo2384636101374064866et_int )
= ( ! [X4: set_set_int] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_286_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X4: nat] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_287_empty__Collect__eq,axiom,
! [P: set_int > $o] :
( ( bot_bot_set_set_int
= ( collect_set_int @ P ) )
= ( ! [X4: set_int] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_288_empty__Collect__eq,axiom,
! [P: set_set_int > $o] :
( ( bot_bo2384636101374064866et_int
= ( collect_set_set_int @ P ) )
= ( ! [X4: set_set_int] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_289_insertCI,axiom,
! [A: set_int,B3: set_set_int,B: set_int] :
( ( ~ ( member_set_int @ A @ B3 )
=> ( A = B ) )
=> ( member_set_int @ A @ ( insert_set_int @ B @ B3 ) ) ) ).
% insertCI
thf(fact_290_insertCI,axiom,
! [A: nat > set_int,B3: set_nat_set_int,B: nat > set_int] :
( ( ~ ( member_nat_set_int @ A @ B3 )
=> ( A = B ) )
=> ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ B3 ) ) ) ).
% insertCI
thf(fact_291_insertCI,axiom,
! [A: nat,B3: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B3 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B3 ) ) ) ).
% insertCI
thf(fact_292_insertCI,axiom,
! [A: set_int > nat,B3: set_set_int_nat,B: set_int > nat] :
( ( ~ ( member_set_int_nat @ A @ B3 )
=> ( A = B ) )
=> ( member_set_int_nat @ A @ ( insert_set_int_nat @ B @ B3 ) ) ) ).
% insertCI
thf(fact_293_insertCI,axiom,
! [A: set_set_int,B3: set_set_set_int,B: set_set_int] :
( ( ~ ( member_set_set_int @ A @ B3 )
=> ( A = B ) )
=> ( member_set_set_int @ A @ ( insert_set_set_int @ B @ B3 ) ) ) ).
% insertCI
thf(fact_294_insert__iff,axiom,
! [A: set_int,B: set_int,A2: set_set_int] :
( ( member_set_int @ A @ ( insert_set_int @ B @ A2 ) )
= ( ( A = B )
| ( member_set_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_295_insert__iff,axiom,
! [A: nat > set_int,B: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_set_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_296_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_297_insert__iff,axiom,
! [A: set_int > nat,B: set_int > nat,A2: set_set_int_nat] :
( ( member_set_int_nat @ A @ ( insert_set_int_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_set_int_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_298_insert__iff,axiom,
! [A: set_set_int,B: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ A @ ( insert_set_set_int @ B @ A2 ) )
= ( ( A = B )
| ( member_set_set_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_299_insert__absorb2,axiom,
! [X2: set_int,A2: set_set_int] :
( ( insert_set_int @ X2 @ ( insert_set_int @ X2 @ A2 ) )
= ( insert_set_int @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_300_insert__absorb2,axiom,
! [X2: set_set_int,A2: set_set_set_int] :
( ( insert_set_set_int @ X2 @ ( insert_set_set_int @ X2 @ A2 ) )
= ( insert_set_set_int @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_301_n__ge__1,axiom,
ord_less_nat @ one_one_nat @ n ).
% n_ge_1
thf(fact_302_subset__empty,axiom,
! [A2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ bot_bot_set_set_int )
= ( A2 = bot_bot_set_set_int ) ) ).
% subset_empty
thf(fact_303_subset__empty,axiom,
! [A2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ bot_bo2384636101374064866et_int )
= ( A2 = bot_bo2384636101374064866et_int ) ) ).
% subset_empty
thf(fact_304_empty__subsetI,axiom,
! [A2: set_set_int] : ( ord_le4403425263959731960et_int @ bot_bot_set_set_int @ A2 ) ).
% empty_subsetI
thf(fact_305_empty__subsetI,axiom,
! [A2: set_set_set_int] : ( ord_le4317611570275147438et_int @ bot_bo2384636101374064866et_int @ A2 ) ).
% empty_subsetI
thf(fact_306_insert__subset,axiom,
! [X2: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ ( insert_nat_set_int @ X2 @ A2 ) @ B3 )
= ( ( member_nat_set_int @ X2 @ B3 )
& ( ord_le5995675665013768039et_int @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_307_insert__subset,axiom,
! [X2: nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A2 ) @ B3 )
= ( ( member_nat @ X2 @ B3 )
& ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_308_insert__subset,axiom,
! [X2: set_int > nat,A2: set_set_int_nat,B3: set_set_int_nat] :
( ( ord_le8005464852987597927nt_nat @ ( insert_set_int_nat @ X2 @ A2 ) @ B3 )
= ( ( member_set_int_nat @ X2 @ B3 )
& ( ord_le8005464852987597927nt_nat @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_309_insert__subset,axiom,
! [X2: set_int,A2: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( insert_set_int @ X2 @ A2 ) @ B3 )
= ( ( member_set_int @ X2 @ B3 )
& ( ord_le4403425263959731960et_int @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_310_insert__subset,axiom,
! [X2: set_set_int,A2: set_set_set_int,B3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ ( insert_set_set_int @ X2 @ A2 ) @ B3 )
= ( ( member_set_set_int @ X2 @ B3 )
& ( ord_le4317611570275147438et_int @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_311_singletonI,axiom,
! [A: nat > set_int] : ( member_nat_set_int @ A @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ) ).
% singletonI
thf(fact_312_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_313_singletonI,axiom,
! [A: set_int > nat] : ( member_set_int_nat @ A @ ( insert_set_int_nat @ A @ bot_bo1204028561185317019nt_nat ) ) ).
% singletonI
thf(fact_314_singletonI,axiom,
! [A: set_int] : ( member_set_int @ A @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ).
% singletonI
thf(fact_315_singletonI,axiom,
! [A: set_set_int] : ( member_set_set_int @ A @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) ).
% singletonI
thf(fact_316_bound_Ointro,axiom,
! [N: nat,F: nat > set_int,Z: set_int] :
( ! [M3: nat] :
( ( ord_less_nat @ N @ M3 )
=> ( ( F @ M3 )
= Z ) )
=> ( bound_set_int @ Z @ N @ F ) ) ).
% bound.intro
thf(fact_317_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_318_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_319_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_320_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_321_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_322_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_323_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_324_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_325_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_326_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_327_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_328_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_329_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_330_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_331_bound_Obound,axiom,
! [Z: set_int,N: nat,F: nat > set_int,M: nat] :
( ( bound_set_int @ Z @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( ( F @ M )
= Z ) ) ) ).
% bound.bound
thf(fact_332_bound__def,axiom,
( bound_set_int
= ( ^ [Z4: set_int,N4: nat,F4: nat > set_int] :
! [M5: nat] :
( ( ord_less_nat @ N4 @ M5 )
=> ( ( F4 @ M5 )
= Z4 ) ) ) ) ).
% bound_def
thf(fact_333_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_334_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_335_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_336_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_337_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_338_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J2: nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_339_mod__ring__carr,axiom,
! [X2: nat,N: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ ( mod_ring @ N ) ) )
= ( ord_less_nat @ X2 @ N ) ) ).
% mod_ring_carr
thf(fact_340_in__mono,axiom,
! [A2: set_nat_set_int,B3: set_nat_set_int,X2: nat > set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ B3 )
=> ( ( member_nat_set_int @ X2 @ A2 )
=> ( member_nat_set_int @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_341_in__mono,axiom,
! [A2: set_nat,B3: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_342_in__mono,axiom,
! [A2: set_set_int_nat,B3: set_set_int_nat,X2: set_int > nat] :
( ( ord_le8005464852987597927nt_nat @ A2 @ B3 )
=> ( ( member_set_int_nat @ X2 @ A2 )
=> ( member_set_int_nat @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_343_in__mono,axiom,
! [A2: set_set_int,B3: set_set_int,X2: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ( ( member_set_int @ X2 @ A2 )
=> ( member_set_int @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_344_in__mono,axiom,
! [A2: set_set_set_int,B3: set_set_set_int,X2: set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ( member_set_set_int @ X2 @ A2 )
=> ( member_set_set_int @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_345_subsetD,axiom,
! [A2: set_nat_set_int,B3: set_nat_set_int,C: nat > set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ B3 )
=> ( ( member_nat_set_int @ C @ A2 )
=> ( member_nat_set_int @ C @ B3 ) ) ) ).
% subsetD
thf(fact_346_subsetD,axiom,
! [A2: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_347_subsetD,axiom,
! [A2: set_set_int_nat,B3: set_set_int_nat,C: set_int > nat] :
( ( ord_le8005464852987597927nt_nat @ A2 @ B3 )
=> ( ( member_set_int_nat @ C @ A2 )
=> ( member_set_int_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_348_subsetD,axiom,
! [A2: set_set_int,B3: set_set_int,C: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ( ( member_set_int @ C @ A2 )
=> ( member_set_int @ C @ B3 ) ) ) ).
% subsetD
thf(fact_349_subsetD,axiom,
! [A2: set_set_set_int,B3: set_set_set_int,C: set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ( member_set_set_int @ C @ A2 )
=> ( member_set_set_int @ C @ B3 ) ) ) ).
% subsetD
thf(fact_350_equalityE,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( A2 = B3 )
=> ~ ( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ~ ( ord_le4403425263959731960et_int @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_351_equalityE,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ( A2 = B3 )
=> ~ ( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ~ ( ord_le4317611570275147438et_int @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_352_subset__eq,axiom,
( ord_le5995675665013768039et_int
= ( ^ [A4: set_nat_set_int,B4: set_nat_set_int] :
! [X4: nat > set_int] :
( ( member_nat_set_int @ X4 @ A4 )
=> ( member_nat_set_int @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_353_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( member_nat @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_354_subset__eq,axiom,
( ord_le8005464852987597927nt_nat
= ( ^ [A4: set_set_int_nat,B4: set_set_int_nat] :
! [X4: set_int > nat] :
( ( member_set_int_nat @ X4 @ A4 )
=> ( member_set_int_nat @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_355_subset__eq,axiom,
( ord_le4403425263959731960et_int
= ( ^ [A4: set_set_int,B4: set_set_int] :
! [X4: set_int] :
( ( member_set_int @ X4 @ A4 )
=> ( member_set_int @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_356_subset__eq,axiom,
( ord_le4317611570275147438et_int
= ( ^ [A4: set_set_set_int,B4: set_set_set_int] :
! [X4: set_set_int] :
( ( member_set_set_int @ X4 @ A4 )
=> ( member_set_set_int @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_357_equalityD1,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( A2 = B3 )
=> ( ord_le4403425263959731960et_int @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_358_equalityD1,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ( A2 = B3 )
=> ( ord_le4317611570275147438et_int @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_359_equalityD2,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( A2 = B3 )
=> ( ord_le4403425263959731960et_int @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_360_equalityD2,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ( A2 = B3 )
=> ( ord_le4317611570275147438et_int @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_361_subset__iff,axiom,
( ord_le5995675665013768039et_int
= ( ^ [A4: set_nat_set_int,B4: set_nat_set_int] :
! [T2: nat > set_int] :
( ( member_nat_set_int @ T2 @ A4 )
=> ( member_nat_set_int @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_362_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A4 )
=> ( member_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_363_subset__iff,axiom,
( ord_le8005464852987597927nt_nat
= ( ^ [A4: set_set_int_nat,B4: set_set_int_nat] :
! [T2: set_int > nat] :
( ( member_set_int_nat @ T2 @ A4 )
=> ( member_set_int_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_364_subset__iff,axiom,
( ord_le4403425263959731960et_int
= ( ^ [A4: set_set_int,B4: set_set_int] :
! [T2: set_int] :
( ( member_set_int @ T2 @ A4 )
=> ( member_set_int @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_365_subset__iff,axiom,
( ord_le4317611570275147438et_int
= ( ^ [A4: set_set_set_int,B4: set_set_set_int] :
! [T2: set_set_int] :
( ( member_set_set_int @ T2 @ A4 )
=> ( member_set_set_int @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_366_subset__refl,axiom,
! [A2: set_set_int] : ( ord_le4403425263959731960et_int @ A2 @ A2 ) ).
% subset_refl
thf(fact_367_subset__refl,axiom,
! [A2: set_set_set_int] : ( ord_le4317611570275147438et_int @ A2 @ A2 ) ).
% subset_refl
thf(fact_368_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_369_Collect__mono,axiom,
! [P: set_int > $o,Q: set_int > $o] :
( ! [X: set_int] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le4403425263959731960et_int @ ( collect_set_int @ P ) @ ( collect_set_int @ Q ) ) ) ).
% Collect_mono
thf(fact_370_Collect__mono,axiom,
! [P: set_set_int > $o,Q: set_set_int > $o] :
( ! [X: set_set_int] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le4317611570275147438et_int @ ( collect_set_set_int @ P ) @ ( collect_set_set_int @ Q ) ) ) ).
% Collect_mono
thf(fact_371_subset__trans,axiom,
! [A2: set_set_int,B3: set_set_int,C2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ( ( ord_le4403425263959731960et_int @ B3 @ C2 )
=> ( ord_le4403425263959731960et_int @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_372_subset__trans,axiom,
! [A2: set_set_set_int,B3: set_set_set_int,C2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ( ord_le4317611570275147438et_int @ B3 @ C2 )
=> ( ord_le4317611570275147438et_int @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_373_set__eq__subset,axiom,
( ( ^ [Y5: set_set_int,Z3: set_set_int] : ( Y5 = Z3 ) )
= ( ^ [A4: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ A4 @ B4 )
& ( ord_le4403425263959731960et_int @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_374_set__eq__subset,axiom,
( ( ^ [Y5: set_set_set_int,Z3: set_set_set_int] : ( Y5 = Z3 ) )
= ( ^ [A4: set_set_set_int,B4: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A4 @ B4 )
& ( ord_le4317611570275147438et_int @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_375_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_376_Collect__mono__iff,axiom,
! [P: set_int > $o,Q: set_int > $o] :
( ( ord_le4403425263959731960et_int @ ( collect_set_int @ P ) @ ( collect_set_int @ Q ) )
= ( ! [X4: set_int] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_377_Collect__mono__iff,axiom,
! [P: set_set_int > $o,Q: set_set_int > $o] :
( ( ord_le4317611570275147438et_int @ ( collect_set_set_int @ P ) @ ( collect_set_set_int @ Q ) )
= ( ! [X4: set_set_int] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_378_abelian__monoid_OboundD__carrier,axiom,
! [G: partia4934656038542163276t_unit,N: nat,F: nat > set_int,M: nat] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( bound_set_int @ ( zero_s6269048424454532197t_unit @ G ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_set_int @ ( F @ M ) @ ( partia966996272515721803t_unit @ G ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_379_abelian__monoid_OboundD__carrier,axiom,
! [G: partia4692342223508353374t_unit,N: nat,F: nat > nat,M: nat] :
( ( abelia362511065248671243t_unit @ G )
=> ( ( bound_nat @ ( zero_n5149899317435570679t_unit @ G ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_nat @ ( F @ M ) @ ( partia3499330772048238685t_unit @ G ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_380_emptyE,axiom,
! [A: nat > set_int] :
~ ( member_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ).
% emptyE
thf(fact_381_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_382_emptyE,axiom,
! [A: set_int > nat] :
~ ( member_set_int_nat @ A @ bot_bo1204028561185317019nt_nat ) ).
% emptyE
thf(fact_383_emptyE,axiom,
! [A: set_int] :
~ ( member_set_int @ A @ bot_bot_set_set_int ) ).
% emptyE
thf(fact_384_emptyE,axiom,
! [A: set_set_int] :
~ ( member_set_set_int @ A @ bot_bo2384636101374064866et_int ) ).
% emptyE
thf(fact_385_equals0D,axiom,
! [A2: set_nat_set_int,A: nat > set_int] :
( ( A2 = bot_bo8417611410066262939et_int )
=> ~ ( member_nat_set_int @ A @ A2 ) ) ).
% equals0D
thf(fact_386_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_387_equals0D,axiom,
! [A2: set_set_int_nat,A: set_int > nat] :
( ( A2 = bot_bo1204028561185317019nt_nat )
=> ~ ( member_set_int_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_388_equals0D,axiom,
! [A2: set_set_int,A: set_int] :
( ( A2 = bot_bot_set_set_int )
=> ~ ( member_set_int @ A @ A2 ) ) ).
% equals0D
thf(fact_389_equals0D,axiom,
! [A2: set_set_set_int,A: set_set_int] :
( ( A2 = bot_bo2384636101374064866et_int )
=> ~ ( member_set_set_int @ A @ A2 ) ) ).
% equals0D
thf(fact_390_equals0I,axiom,
! [A2: set_nat_set_int] :
( ! [Y3: nat > set_int] :
~ ( member_nat_set_int @ Y3 @ A2 )
=> ( A2 = bot_bo8417611410066262939et_int ) ) ).
% equals0I
thf(fact_391_equals0I,axiom,
! [A2: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_392_equals0I,axiom,
! [A2: set_set_int_nat] :
( ! [Y3: set_int > nat] :
~ ( member_set_int_nat @ Y3 @ A2 )
=> ( A2 = bot_bo1204028561185317019nt_nat ) ) ).
% equals0I
thf(fact_393_equals0I,axiom,
! [A2: set_set_int] :
( ! [Y3: set_int] :
~ ( member_set_int @ Y3 @ A2 )
=> ( A2 = bot_bot_set_set_int ) ) ).
% equals0I
thf(fact_394_equals0I,axiom,
! [A2: set_set_set_int] :
( ! [Y3: set_set_int] :
~ ( member_set_set_int @ Y3 @ A2 )
=> ( A2 = bot_bo2384636101374064866et_int ) ) ).
% equals0I
thf(fact_395_ex__in__conv,axiom,
! [A2: set_nat_set_int] :
( ( ? [X4: nat > set_int] : ( member_nat_set_int @ X4 @ A2 ) )
= ( A2 != bot_bo8417611410066262939et_int ) ) ).
% ex_in_conv
thf(fact_396_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X4: nat] : ( member_nat @ X4 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_397_ex__in__conv,axiom,
! [A2: set_set_int_nat] :
( ( ? [X4: set_int > nat] : ( member_set_int_nat @ X4 @ A2 ) )
= ( A2 != bot_bo1204028561185317019nt_nat ) ) ).
% ex_in_conv
thf(fact_398_ex__in__conv,axiom,
! [A2: set_set_int] :
( ( ? [X4: set_int] : ( member_set_int @ X4 @ A2 ) )
= ( A2 != bot_bot_set_set_int ) ) ).
% ex_in_conv
thf(fact_399_ex__in__conv,axiom,
! [A2: set_set_set_int] :
( ( ? [X4: set_set_int] : ( member_set_set_int @ X4 @ A2 ) )
= ( A2 != bot_bo2384636101374064866et_int ) ) ).
% ex_in_conv
thf(fact_400_insertE,axiom,
! [A: set_int,B: set_int,A2: set_set_int] :
( ( member_set_int @ A @ ( insert_set_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_401_insertE,axiom,
! [A: nat > set_int,B: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat_set_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_402_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_403_insertE,axiom,
! [A: set_int > nat,B: set_int > nat,A2: set_set_int_nat] :
( ( member_set_int_nat @ A @ ( insert_set_int_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_int_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_404_insertE,axiom,
! [A: set_set_int,B: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ A @ ( insert_set_set_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_set_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_405_insertI1,axiom,
! [A: set_int,B3: set_set_int] : ( member_set_int @ A @ ( insert_set_int @ A @ B3 ) ) ).
% insertI1
thf(fact_406_insertI1,axiom,
! [A: nat > set_int,B3: set_nat_set_int] : ( member_nat_set_int @ A @ ( insert_nat_set_int @ A @ B3 ) ) ).
% insertI1
thf(fact_407_insertI1,axiom,
! [A: nat,B3: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B3 ) ) ).
% insertI1
thf(fact_408_insertI1,axiom,
! [A: set_int > nat,B3: set_set_int_nat] : ( member_set_int_nat @ A @ ( insert_set_int_nat @ A @ B3 ) ) ).
% insertI1
thf(fact_409_insertI1,axiom,
! [A: set_set_int,B3: set_set_set_int] : ( member_set_set_int @ A @ ( insert_set_set_int @ A @ B3 ) ) ).
% insertI1
thf(fact_410_insertI2,axiom,
! [A: set_int,B3: set_set_int,B: set_int] :
( ( member_set_int @ A @ B3 )
=> ( member_set_int @ A @ ( insert_set_int @ B @ B3 ) ) ) ).
% insertI2
thf(fact_411_insertI2,axiom,
! [A: nat > set_int,B3: set_nat_set_int,B: nat > set_int] :
( ( member_nat_set_int @ A @ B3 )
=> ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ B3 ) ) ) ).
% insertI2
thf(fact_412_insertI2,axiom,
! [A: nat,B3: set_nat,B: nat] :
( ( member_nat @ A @ B3 )
=> ( member_nat @ A @ ( insert_nat @ B @ B3 ) ) ) ).
% insertI2
thf(fact_413_insertI2,axiom,
! [A: set_int > nat,B3: set_set_int_nat,B: set_int > nat] :
( ( member_set_int_nat @ A @ B3 )
=> ( member_set_int_nat @ A @ ( insert_set_int_nat @ B @ B3 ) ) ) ).
% insertI2
thf(fact_414_insertI2,axiom,
! [A: set_set_int,B3: set_set_set_int,B: set_set_int] :
( ( member_set_set_int @ A @ B3 )
=> ( member_set_set_int @ A @ ( insert_set_set_int @ B @ B3 ) ) ) ).
% insertI2
thf(fact_415_Set_Oset__insert,axiom,
! [X2: set_int,A2: set_set_int] :
( ( member_set_int @ X2 @ A2 )
=> ~ ! [B5: set_set_int] :
( ( A2
= ( insert_set_int @ X2 @ B5 ) )
=> ( member_set_int @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_416_Set_Oset__insert,axiom,
! [X2: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ X2 @ A2 )
=> ~ ! [B5: set_nat_set_int] :
( ( A2
= ( insert_nat_set_int @ X2 @ B5 ) )
=> ( member_nat_set_int @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_417_Set_Oset__insert,axiom,
! [X2: nat,A2: set_nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ! [B5: set_nat] :
( ( A2
= ( insert_nat @ X2 @ B5 ) )
=> ( member_nat @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_418_Set_Oset__insert,axiom,
! [X2: set_int > nat,A2: set_set_int_nat] :
( ( member_set_int_nat @ X2 @ A2 )
=> ~ ! [B5: set_set_int_nat] :
( ( A2
= ( insert_set_int_nat @ X2 @ B5 ) )
=> ( member_set_int_nat @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_419_Set_Oset__insert,axiom,
! [X2: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ X2 @ A2 )
=> ~ ! [B5: set_set_set_int] :
( ( A2
= ( insert_set_set_int @ X2 @ B5 ) )
=> ( member_set_set_int @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_420_insert__ident,axiom,
! [X2: set_int,A2: set_set_int,B3: set_set_int] :
( ~ ( member_set_int @ X2 @ A2 )
=> ( ~ ( member_set_int @ X2 @ B3 )
=> ( ( ( insert_set_int @ X2 @ A2 )
= ( insert_set_int @ X2 @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_421_insert__ident,axiom,
! [X2: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ~ ( member_nat_set_int @ X2 @ A2 )
=> ( ~ ( member_nat_set_int @ X2 @ B3 )
=> ( ( ( insert_nat_set_int @ X2 @ A2 )
= ( insert_nat_set_int @ X2 @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_422_insert__ident,axiom,
! [X2: nat,A2: set_nat,B3: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ~ ( member_nat @ X2 @ B3 )
=> ( ( ( insert_nat @ X2 @ A2 )
= ( insert_nat @ X2 @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_423_insert__ident,axiom,
! [X2: set_int > nat,A2: set_set_int_nat,B3: set_set_int_nat] :
( ~ ( member_set_int_nat @ X2 @ A2 )
=> ( ~ ( member_set_int_nat @ X2 @ B3 )
=> ( ( ( insert_set_int_nat @ X2 @ A2 )
= ( insert_set_int_nat @ X2 @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_424_insert__ident,axiom,
! [X2: set_set_int,A2: set_set_set_int,B3: set_set_set_int] :
( ~ ( member_set_set_int @ X2 @ A2 )
=> ( ~ ( member_set_set_int @ X2 @ B3 )
=> ( ( ( insert_set_set_int @ X2 @ A2 )
= ( insert_set_set_int @ X2 @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_425_insert__absorb,axiom,
! [A: set_int,A2: set_set_int] :
( ( member_set_int @ A @ A2 )
=> ( ( insert_set_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_426_insert__absorb,axiom,
! [A: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ A2 )
=> ( ( insert_nat_set_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_427_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_428_insert__absorb,axiom,
! [A: set_int > nat,A2: set_set_int_nat] :
( ( member_set_int_nat @ A @ A2 )
=> ( ( insert_set_int_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_429_insert__absorb,axiom,
! [A: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ A @ A2 )
=> ( ( insert_set_set_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_430_insert__eq__iff,axiom,
! [A: set_int,A2: set_set_int,B: set_int,B3: set_set_int] :
( ~ ( member_set_int @ A @ A2 )
=> ( ~ ( member_set_int @ B @ B3 )
=> ( ( ( insert_set_int @ A @ A2 )
= ( insert_set_int @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C3: set_set_int] :
( ( A2
= ( insert_set_int @ B @ C3 ) )
& ~ ( member_set_int @ B @ C3 )
& ( B3
= ( insert_set_int @ A @ C3 ) )
& ~ ( member_set_int @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_431_insert__eq__iff,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B: nat > set_int,B3: set_nat_set_int] :
( ~ ( member_nat_set_int @ A @ A2 )
=> ( ~ ( member_nat_set_int @ B @ B3 )
=> ( ( ( insert_nat_set_int @ A @ A2 )
= ( insert_nat_set_int @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C3: set_nat_set_int] :
( ( A2
= ( insert_nat_set_int @ B @ C3 ) )
& ~ ( member_nat_set_int @ B @ C3 )
& ( B3
= ( insert_nat_set_int @ A @ C3 ) )
& ~ ( member_nat_set_int @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_432_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B3: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B3 )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B3
= ( insert_nat @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_433_insert__eq__iff,axiom,
! [A: set_int > nat,A2: set_set_int_nat,B: set_int > nat,B3: set_set_int_nat] :
( ~ ( member_set_int_nat @ A @ A2 )
=> ( ~ ( member_set_int_nat @ B @ B3 )
=> ( ( ( insert_set_int_nat @ A @ A2 )
= ( insert_set_int_nat @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C3: set_set_int_nat] :
( ( A2
= ( insert_set_int_nat @ B @ C3 ) )
& ~ ( member_set_int_nat @ B @ C3 )
& ( B3
= ( insert_set_int_nat @ A @ C3 ) )
& ~ ( member_set_int_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_434_insert__eq__iff,axiom,
! [A: set_set_int,A2: set_set_set_int,B: set_set_int,B3: set_set_set_int] :
( ~ ( member_set_set_int @ A @ A2 )
=> ( ~ ( member_set_set_int @ B @ B3 )
=> ( ( ( insert_set_set_int @ A @ A2 )
= ( insert_set_set_int @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C3: set_set_set_int] :
( ( A2
= ( insert_set_set_int @ B @ C3 ) )
& ~ ( member_set_set_int @ B @ C3 )
& ( B3
= ( insert_set_set_int @ A @ C3 ) )
& ~ ( member_set_set_int @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_435_insert__commute,axiom,
! [X2: set_int,Y: set_int,A2: set_set_int] :
( ( insert_set_int @ X2 @ ( insert_set_int @ Y @ A2 ) )
= ( insert_set_int @ Y @ ( insert_set_int @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_436_insert__commute,axiom,
! [X2: set_set_int,Y: set_set_int,A2: set_set_set_int] :
( ( insert_set_set_int @ X2 @ ( insert_set_set_int @ Y @ A2 ) )
= ( insert_set_set_int @ Y @ ( insert_set_set_int @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_437_mk__disjoint__insert,axiom,
! [A: set_int,A2: set_set_int] :
( ( member_set_int @ A @ A2 )
=> ? [B5: set_set_int] :
( ( A2
= ( insert_set_int @ A @ B5 ) )
& ~ ( member_set_int @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_438_mk__disjoint__insert,axiom,
! [A: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ A2 )
=> ? [B5: set_nat_set_int] :
( ( A2
= ( insert_nat_set_int @ A @ B5 ) )
& ~ ( member_nat_set_int @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_439_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B5: set_nat] :
( ( A2
= ( insert_nat @ A @ B5 ) )
& ~ ( member_nat @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_440_mk__disjoint__insert,axiom,
! [A: set_int > nat,A2: set_set_int_nat] :
( ( member_set_int_nat @ A @ A2 )
=> ? [B5: set_set_int_nat] :
( ( A2
= ( insert_set_int_nat @ A @ B5 ) )
& ~ ( member_set_int_nat @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_441_mk__disjoint__insert,axiom,
! [A: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ A @ A2 )
=> ? [B5: set_set_set_int] :
( ( A2
= ( insert_set_set_int @ A @ B5 ) )
& ~ ( member_set_set_int @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_442_principalideal_Ois__principalideal,axiom,
! [I3: set_set_int,R3: partia4934656038542163276t_unit] :
( ( princi8860937869964495385t_unit @ I3 @ R3 )
=> ( princi8860937869964495385t_unit @ I3 @ R3 ) ) ).
% principalideal.is_principalideal
thf(fact_443_Collect__subset,axiom,
! [A2: set_nat_set_int,P: ( nat > set_int ) > $o] :
( ord_le5995675665013768039et_int
@ ( collect_nat_set_int
@ ^ [X4: nat > set_int] :
( ( member_nat_set_int @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_444_Collect__subset,axiom,
! [A2: set_set_int_nat,P: ( set_int > nat ) > $o] :
( ord_le8005464852987597927nt_nat
@ ( collect_set_int_nat
@ ^ [X4: set_int > nat] :
( ( member_set_int_nat @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_445_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_446_Collect__subset,axiom,
! [A2: set_set_int,P: set_int > $o] :
( ord_le4403425263959731960et_int
@ ( collect_set_int
@ ^ [X4: set_int] :
( ( member_set_int @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_447_Collect__subset,axiom,
! [A2: set_set_set_int,P: set_set_int > $o] :
( ord_le4317611570275147438et_int
@ ( collect_set_set_int
@ ^ [X4: set_set_int] :
( ( member_set_set_int @ X4 @ A2 )
& ( P @ X4 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_448_less__eq__set__def,axiom,
( ord_le5995675665013768039et_int
= ( ^ [A4: set_nat_set_int,B4: set_nat_set_int] :
( ord_le8937012796915117022_int_o
@ ^ [X4: nat > set_int] : ( member_nat_set_int @ X4 @ A4 )
@ ^ [X4: nat > set_int] : ( member_nat_set_int @ X4 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_449_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_450_less__eq__set__def,axiom,
( ord_le8005464852987597927nt_nat
= ( ^ [A4: set_set_int_nat,B4: set_set_int_nat] :
( ord_le2186727632418068190_nat_o
@ ^ [X4: set_int > nat] : ( member_set_int_nat @ X4 @ A4 )
@ ^ [X4: set_int > nat] : ( member_set_int_nat @ X4 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_451_less__eq__set__def,axiom,
( ord_le4403425263959731960et_int
= ( ^ [A4: set_set_int,B4: set_set_int] :
( ord_le1719207048323850125_int_o
@ ^ [X4: set_int] : ( member_set_int @ X4 @ A4 )
@ ^ [X4: set_int] : ( member_set_int @ X4 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_452_less__eq__set__def,axiom,
( ord_le4317611570275147438et_int
= ( ^ [A4: set_set_set_int,B4: set_set_set_int] :
( ord_le3688374355499189079_int_o
@ ^ [X4: set_set_int] : ( member_set_set_int @ X4 @ A4 )
@ ^ [X4: set_set_int] : ( member_set_set_int @ X4 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_453_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X4: nat] : $false ) ) ).
% empty_def
thf(fact_454_empty__def,axiom,
( bot_bot_set_set_int
= ( collect_set_int
@ ^ [X4: set_int] : $false ) ) ).
% empty_def
thf(fact_455_empty__def,axiom,
( bot_bo2384636101374064866et_int
= ( collect_set_set_int
@ ^ [X4: set_set_int] : $false ) ) ).
% empty_def
thf(fact_456_insert__compr,axiom,
( insert_set_int
= ( ^ [A5: set_int,B4: set_set_int] :
( collect_set_int
@ ^ [X4: set_int] :
( ( X4 = A5 )
| ( member_set_int @ X4 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_457_insert__compr,axiom,
( insert_nat_set_int
= ( ^ [A5: nat > set_int,B4: set_nat_set_int] :
( collect_nat_set_int
@ ^ [X4: nat > set_int] :
( ( X4 = A5 )
| ( member_nat_set_int @ X4 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_458_insert__compr,axiom,
( insert_set_int_nat
= ( ^ [A5: set_int > nat,B4: set_set_int_nat] :
( collect_set_int_nat
@ ^ [X4: set_int > nat] :
( ( X4 = A5 )
| ( member_set_int_nat @ X4 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_459_insert__compr,axiom,
( insert_nat
= ( ^ [A5: nat,B4: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( X4 = A5 )
| ( member_nat @ X4 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_460_insert__compr,axiom,
( insert_set_set_int
= ( ^ [A5: set_set_int,B4: set_set_set_int] :
( collect_set_set_int
@ ^ [X4: set_set_int] :
( ( X4 = A5 )
| ( member_set_set_int @ X4 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_461_insert__Collect,axiom,
! [A: set_int,P: set_int > $o] :
( ( insert_set_int @ A @ ( collect_set_int @ P ) )
= ( collect_set_int
@ ^ [U2: set_int] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_462_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_463_insert__Collect,axiom,
! [A: set_set_int,P: set_set_int > $o] :
( ( insert_set_set_int @ A @ ( collect_set_set_int @ P ) )
= ( collect_set_set_int
@ ^ [U2: set_set_int] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_464_subset__insertI2,axiom,
! [A2: set_set_int,B3: set_set_int,B: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_465_subset__insertI2,axiom,
! [A2: set_set_set_int,B3: set_set_set_int,B: set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_466_subset__insertI,axiom,
! [B3: set_set_int,A: set_int] : ( ord_le4403425263959731960et_int @ B3 @ ( insert_set_int @ A @ B3 ) ) ).
% subset_insertI
thf(fact_467_subset__insertI,axiom,
! [B3: set_set_set_int,A: set_set_int] : ( ord_le4317611570275147438et_int @ B3 @ ( insert_set_set_int @ A @ B3 ) ) ).
% subset_insertI
thf(fact_468_subset__insert,axiom,
! [X2: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ~ ( member_nat_set_int @ X2 @ A2 )
=> ( ( ord_le5995675665013768039et_int @ A2 @ ( insert_nat_set_int @ X2 @ B3 ) )
= ( ord_le5995675665013768039et_int @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_469_subset__insert,axiom,
! [X2: nat,A2: set_nat,B3: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B3 ) )
= ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_470_subset__insert,axiom,
! [X2: set_int > nat,A2: set_set_int_nat,B3: set_set_int_nat] :
( ~ ( member_set_int_nat @ X2 @ A2 )
=> ( ( ord_le8005464852987597927nt_nat @ A2 @ ( insert_set_int_nat @ X2 @ B3 ) )
= ( ord_le8005464852987597927nt_nat @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_471_subset__insert,axiom,
! [X2: set_int,A2: set_set_int,B3: set_set_int] :
( ~ ( member_set_int @ X2 @ A2 )
=> ( ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X2 @ B3 ) )
= ( ord_le4403425263959731960et_int @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_472_subset__insert,axiom,
! [X2: set_set_int,A2: set_set_set_int,B3: set_set_set_int] :
( ~ ( member_set_set_int @ X2 @ A2 )
=> ( ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ X2 @ B3 ) )
= ( ord_le4317611570275147438et_int @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_473_insert__mono,axiom,
! [C2: set_set_int,D: set_set_int,A: set_int] :
( ( ord_le4403425263959731960et_int @ C2 @ D )
=> ( ord_le4403425263959731960et_int @ ( insert_set_int @ A @ C2 ) @ ( insert_set_int @ A @ D ) ) ) ).
% insert_mono
thf(fact_474_insert__mono,axiom,
! [C2: set_set_set_int,D: set_set_set_int,A: set_set_int] :
( ( ord_le4317611570275147438et_int @ C2 @ D )
=> ( ord_le4317611570275147438et_int @ ( insert_set_set_int @ A @ C2 ) @ ( insert_set_set_int @ A @ D ) ) ) ).
% insert_mono
thf(fact_475_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_476_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_477_singletonD,axiom,
! [B: set_int > nat,A: set_int > nat] :
( ( member_set_int_nat @ B @ ( insert_set_int_nat @ A @ bot_bo1204028561185317019nt_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_478_singletonD,axiom,
! [B: set_int,A: set_int] :
( ( member_set_int @ B @ ( insert_set_int @ A @ bot_bot_set_set_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_479_singletonD,axiom,
! [B: set_set_int,A: set_set_int] :
( ( member_set_set_int @ B @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_480_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_481_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_482_singleton__iff,axiom,
! [B: set_int > nat,A: set_int > nat] :
( ( member_set_int_nat @ B @ ( insert_set_int_nat @ A @ bot_bo1204028561185317019nt_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_483_singleton__iff,axiom,
! [B: set_int,A: set_int] :
( ( member_set_int @ B @ ( insert_set_int @ A @ bot_bot_set_set_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_484_singleton__iff,axiom,
! [B: set_set_int,A: set_set_int] :
( ( member_set_set_int @ B @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_485_doubleton__eq__iff,axiom,
! [A: set_int,B: set_int,C: set_int,D2: set_int] :
( ( ( insert_set_int @ A @ ( insert_set_int @ B @ bot_bot_set_set_int ) )
= ( insert_set_int @ C @ ( insert_set_int @ D2 @ bot_bot_set_set_int ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_486_doubleton__eq__iff,axiom,
! [A: set_set_int,B: set_set_int,C: set_set_int,D2: set_set_int] :
( ( ( insert_set_set_int @ A @ ( insert_set_set_int @ B @ bot_bo2384636101374064866et_int ) )
= ( insert_set_set_int @ C @ ( insert_set_set_int @ D2 @ bot_bo2384636101374064866et_int ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_487_insert__not__empty,axiom,
! [A: set_int,A2: set_set_int] :
( ( insert_set_int @ A @ A2 )
!= bot_bot_set_set_int ) ).
% insert_not_empty
thf(fact_488_insert__not__empty,axiom,
! [A: set_set_int,A2: set_set_set_int] :
( ( insert_set_set_int @ A @ A2 )
!= bot_bo2384636101374064866et_int ) ).
% insert_not_empty
thf(fact_489_singleton__inject,axiom,
! [A: set_int,B: set_int] :
( ( ( insert_set_int @ A @ bot_bot_set_set_int )
= ( insert_set_int @ B @ bot_bot_set_set_int ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_490_singleton__inject,axiom,
! [A: set_set_int,B: set_set_int] :
( ( ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int )
= ( insert_set_set_int @ B @ bot_bo2384636101374064866et_int ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_491_mem__upD,axiom,
! [F: nat > set_int,R3: partia4934656038542163276t_unit,N: nat] :
( ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ R3 ) )
=> ( member_set_int @ ( F @ N ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ).
% mem_upD
thf(fact_492_mem__upD,axiom,
! [F: nat > nat,R3: partia4692342223508353374t_unit,N: nat] :
( ( member_nat_nat @ F @ ( up_nat_Product_unit @ R3 ) )
=> ( member_nat @ ( F @ N ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ).
% mem_upD
thf(fact_493_bound__below,axiom,
! [Z: set_int,M: nat,F: nat > set_int,N: nat] :
( ( bound_set_int @ Z @ M @ F )
=> ( ( ( F @ N )
!= Z )
=> ( ord_less_eq_nat @ N @ M ) ) ) ).
% bound_below
thf(fact_494_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X4: nat] :
( ( X4 = A )
& ( P @ X4 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X4: nat] :
( ( X4 = A )
& ( P @ X4 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_495_Collect__conv__if,axiom,
! [P: set_int > $o,A: set_int] :
( ( ( P @ A )
=> ( ( collect_set_int
@ ^ [X4: set_int] :
( ( X4 = A )
& ( P @ X4 ) ) )
= ( insert_set_int @ A @ bot_bot_set_set_int ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_int
@ ^ [X4: set_int] :
( ( X4 = A )
& ( P @ X4 ) ) )
= bot_bot_set_set_int ) ) ) ).
% Collect_conv_if
thf(fact_496_Collect__conv__if,axiom,
! [P: set_set_int > $o,A: set_set_int] :
( ( ( P @ A )
=> ( ( collect_set_set_int
@ ^ [X4: set_set_int] :
( ( X4 = A )
& ( P @ X4 ) ) )
= ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_set_int
@ ^ [X4: set_set_int] :
( ( X4 = A )
& ( P @ X4 ) ) )
= bot_bo2384636101374064866et_int ) ) ) ).
% Collect_conv_if
thf(fact_497_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X4: nat] :
( ( A = X4 )
& ( P @ X4 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X4: nat] :
( ( A = X4 )
& ( P @ X4 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_498_Collect__conv__if2,axiom,
! [P: set_int > $o,A: set_int] :
( ( ( P @ A )
=> ( ( collect_set_int
@ ^ [X4: set_int] :
( ( A = X4 )
& ( P @ X4 ) ) )
= ( insert_set_int @ A @ bot_bot_set_set_int ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_int
@ ^ [X4: set_int] :
( ( A = X4 )
& ( P @ X4 ) ) )
= bot_bot_set_set_int ) ) ) ).
% Collect_conv_if2
thf(fact_499_Collect__conv__if2,axiom,
! [P: set_set_int > $o,A: set_set_int] :
( ( ( P @ A )
=> ( ( collect_set_set_int
@ ^ [X4: set_set_int] :
( ( A = X4 )
& ( P @ X4 ) ) )
= ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_set_int
@ ^ [X4: set_set_int] :
( ( A = X4 )
& ( P @ X4 ) ) )
= bot_bo2384636101374064866et_int ) ) ) ).
% Collect_conv_if2
thf(fact_500_subset__singleton__iff,axiom,
! [X5: set_set_int,A: set_int] :
( ( ord_le4403425263959731960et_int @ X5 @ ( insert_set_int @ A @ bot_bot_set_set_int ) )
= ( ( X5 = bot_bot_set_set_int )
| ( X5
= ( insert_set_int @ A @ bot_bot_set_set_int ) ) ) ) ).
% subset_singleton_iff
thf(fact_501_subset__singleton__iff,axiom,
! [X5: set_set_set_int,A: set_set_int] :
( ( ord_le4317611570275147438et_int @ X5 @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) )
= ( ( X5 = bot_bo2384636101374064866et_int )
| ( X5
= ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) ) ) ).
% subset_singleton_iff
thf(fact_502_subset__singletonD,axiom,
! [A2: set_set_int,X2: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X2 @ bot_bot_set_set_int ) )
=> ( ( A2 = bot_bot_set_set_int )
| ( A2
= ( insert_set_int @ X2 @ bot_bot_set_set_int ) ) ) ) ).
% subset_singletonD
thf(fact_503_subset__singletonD,axiom,
! [A2: set_set_set_int,X2: set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ X2 @ bot_bo2384636101374064866et_int ) )
=> ( ( A2 = bot_bo2384636101374064866et_int )
| ( A2
= ( insert_set_set_int @ X2 @ bot_bo2384636101374064866et_int ) ) ) ) ).
% subset_singletonD
thf(fact_504_ring_Ocgenideal__self,axiom,
! [R3: partia4934656038542163276t_unit,I2: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ I2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ I2 @ ( cgenid8502489213727343375t_unit @ R3 @ I2 ) ) ) ) ).
% ring.cgenideal_self
thf(fact_505_ring_Ocgenideal__self,axiom,
! [R3: partia4692342223508353374t_unit,I2: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ I2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ I2 @ ( cgenid8693976350862946099t_unit @ R3 @ I2 ) ) ) ) ).
% ring.cgenideal_self
thf(fact_506_ring_Oonepideal,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( princi8860937869964495385t_unit @ ( partia966996272515721803t_unit @ R3 ) @ R3 ) ) ).
% ring.onepideal
thf(fact_507_ring_Oonepideal,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( princi4652470909602072491t_unit @ ( partia3499330772048238685t_unit @ R3 ) @ R3 ) ) ).
% ring.onepideal
thf(fact_508_ring_Oup__add__closed,axiom,
! [R3: partia4934656038542163276t_unit,P2: nat > set_int,Q2: nat > set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_nat_set_int @ P2 @ ( up_set1168727741560211120t_unit @ R3 ) )
=> ( ( member_nat_set_int @ Q2 @ ( up_set1168727741560211120t_unit @ R3 ) )
=> ( member_nat_set_int
@ ^ [I: nat] : ( add_se5859248395121729892t_unit @ R3 @ ( P2 @ I ) @ ( Q2 @ I ) )
@ ( up_set1168727741560211120t_unit @ R3 ) ) ) ) ) ).
% ring.up_add_closed
thf(fact_509_ring_Oup__minus__closed,axiom,
! [R3: partia4934656038542163276t_unit,P2: nat > set_int,Q2: nat > set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_nat_set_int @ P2 @ ( up_set1168727741560211120t_unit @ R3 ) )
=> ( ( member_nat_set_int @ Q2 @ ( up_set1168727741560211120t_unit @ R3 ) )
=> ( member_nat_set_int
@ ^ [I: nat] : ( a_minu5974516859897376926t_unit @ R3 @ ( P2 @ I ) @ ( Q2 @ I ) )
@ ( up_set1168727741560211120t_unit @ R3 ) ) ) ) ) ).
% ring.up_minus_closed
thf(fact_510_ring_Oset__add__comm,axiom,
! [R3: partia4934656038542163276t_unit,I3: set_set_int,J: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( ord_le4403425263959731960et_int @ I3 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( ord_le4403425263959731960et_int @ J @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( set_ad273131178244904872t_unit @ R3 @ I3 @ J )
= ( set_ad273131178244904872t_unit @ R3 @ J @ I3 ) ) ) ) ) ).
% ring.set_add_comm
thf(fact_511_ring_Oset__add__comm,axiom,
! [R3: partia4692342223508353374t_unit,I3: set_nat,J: set_nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( ord_less_eq_set_nat @ I3 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( ord_less_eq_set_nat @ J @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( set_ad320919470248169786t_unit @ R3 @ I3 @ J )
= ( set_ad320919470248169786t_unit @ R3 @ J @ I3 ) ) ) ) ) ).
% ring.set_add_comm
thf(fact_512_ring_Osubset__Idl__subset,axiom,
! [R3: partia4934656038542163276t_unit,I3: set_set_int,H3: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( ord_le4403425263959731960et_int @ I3 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( ord_le4403425263959731960et_int @ H3 @ I3 )
=> ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ R3 @ H3 ) @ ( genide1545711809618862555t_unit @ R3 @ I3 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_513_ring_Osubset__Idl__subset,axiom,
! [R3: partia4692342223508353374t_unit,I3: set_nat,H3: set_nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( ord_less_eq_set_nat @ I3 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( ord_less_eq_set_nat @ H3 @ I3 )
=> ( ord_less_eq_set_nat @ ( genide4496968333291595629t_unit @ R3 @ H3 ) @ ( genide4496968333291595629t_unit @ R3 @ I3 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_514_ring_Ogenideal__self,axiom,
! [R3: partia4934656038542163276t_unit,S: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( ord_le4403425263959731960et_int @ S @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ord_le4403425263959731960et_int @ S @ ( genide1545711809618862555t_unit @ R3 @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_515_ring_Ogenideal__self,axiom,
! [R3: partia4692342223508353374t_unit,S: set_nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( ord_less_eq_set_nat @ S @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ord_less_eq_set_nat @ S @ ( genide4496968333291595629t_unit @ R3 @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_516_ring_Obound__upD,axiom,
! [R3: partia4934656038542163276t_unit,F: nat > set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ R3 ) )
=> ? [N2: nat] : ( bound_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ N2 @ F ) ) ) ).
% ring.bound_upD
thf(fact_517_cring_Ocgenideal__is__principalideal,axiom,
! [R3: partia4934656038542163276t_unit,I2: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ I2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( princi8860937869964495385t_unit @ ( cgenid8502489213727343375t_unit @ R3 @ I2 ) @ R3 ) ) ) ).
% cring.cgenideal_is_principalideal
thf(fact_518_cring_Ocgenideal__is__principalideal,axiom,
! [R3: partia4692342223508353374t_unit,I2: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ I2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( princi4652470909602072491t_unit @ ( cgenid8693976350862946099t_unit @ R3 @ I2 ) @ R3 ) ) ) ).
% cring.cgenideal_is_principalideal
thf(fact_519_ring_Ogenideal__self_H,axiom,
! [R3: partia4934656038542163276t_unit,I2: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ I2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ I2 @ ( genide1545711809618862555t_unit @ R3 @ ( insert_set_int @ I2 @ bot_bot_set_set_int ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_520_ring_Ogenideal__self_H,axiom,
! [R3: partia4692342223508353374t_unit,I2: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ I2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ I2 @ ( genide4496968333291595629t_unit @ R3 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_521_ring_Ogenideal__zero,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( genide1545711809618862555t_unit @ R3 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ bot_bot_set_set_int ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ bot_bot_set_set_int ) ) ) ).
% ring.genideal_zero
thf(fact_522_ring_Ozeropideal,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( princi8860937869964495385t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ bot_bot_set_set_int ) @ R3 ) ) ).
% ring.zeropideal
thf(fact_523_principalideal_Ogenerate,axiom,
! [I3: set_set_int,R3: partia4934656038542163276t_unit] :
( ( princi8860937869964495385t_unit @ I3 @ R3 )
=> ? [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
& ( I3
= ( genide1545711809618862555t_unit @ R3 @ ( insert_set_int @ X @ bot_bot_set_set_int ) ) ) ) ) ).
% principalideal.generate
thf(fact_524_principalideal_Ogenerate,axiom,
! [I3: set_nat,R3: partia4692342223508353374t_unit] :
( ( princi4652470909602072491t_unit @ I3 @ R3 )
=> ? [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
& ( I3
= ( genide4496968333291595629t_unit @ R3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% principalideal.generate
thf(fact_525_line__extension__in__carrier,axiom,
! [K: set_set_int,A: set_int,E: set_set_int] :
( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ A @ E ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_526_genideal__one,axiom,
( ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
= ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% genideal_one
thf(fact_527_add_Oone__in__subset,axiom,
! [H3: set_set_int] :
( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( H3 != bot_bot_set_set_int )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ H3 )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ H3 ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ H3 )
=> ! [Xa2: set_int] :
( ( member_set_int @ Xa2 @ H3 )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Xa2 ) @ H3 ) ) )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ H3 ) ) ) ) ) ).
% add.one_in_subset
thf(fact_528_ring_Osubalbegra__incl__imp__finite__dimension,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int,E: set_set_int,V3: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( subfie3888952257595785920t_unit @ K @ R3 )
=> ( ( embedd8246663962306818995t_unit @ R3 @ K @ E )
=> ( ( embedd2743979684206749024t_unit @ K @ V3 @ R3 )
=> ( ( ord_le4403425263959731960et_int @ V3 @ E )
=> ( embedd8246663962306818995t_unit @ R3 @ K @ V3 ) ) ) ) ) ) ).
% ring.subalbegra_incl_imp_finite_dimension
thf(fact_529_cring_Oto__contain__is__to__divide,axiom,
! [R3: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( ord_le4403425263959731960et_int @ ( cgenid8502489213727343375t_unit @ R3 @ B ) @ ( cgenid8502489213727343375t_unit @ R3 @ A ) )
= ( factor5186451337065598620t_unit @ R3 @ A @ B ) ) ) ) ) ).
% cring.to_contain_is_to_divide
thf(fact_530_cring_Oto__contain__is__to__divide,axiom,
! [R3: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( ord_less_eq_set_nat @ ( cgenid8693976350862946099t_unit @ R3 @ B ) @ ( cgenid8693976350862946099t_unit @ R3 @ A ) )
= ( factor7017787129603596992t_unit @ R3 @ A @ B ) ) ) ) ) ).
% cring.to_contain_is_to_divide
thf(fact_531_one__zeroI,axiom,
( ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
=> ( ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% one_zeroI
thf(fact_532_one__zeroD,axiom,
( ( ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) ) ).
% one_zeroD
thf(fact_533_carrier__one__zero,axiom,
( ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
= ( ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% carrier_one_zero
thf(fact_534_psubsetI,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less_set_set_int @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_535_psubsetI,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_le4562804192517611682et_int @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_536_subring__props_I5_J,axiom,
! [K: set_set_int,H: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ H @ K )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H ) @ K ) ) ) ).
% subring_props(5)
thf(fact_537_subring__props_I3_J,axiom,
! [K: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ K ) ) ).
% subring_props(3)
thf(fact_538_up__a__inv__closed,axiom,
! [P2: nat > set_int] :
( ( member_nat_set_int @ P2 @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_nat_set_int
@ ^ [I: nat] : ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( P2 @ I ) )
@ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% up_a_inv_closed
thf(fact_539_add_Oinv__mult__group,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_540_add_Oinv__solve__left,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( A
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B ) @ C ) )
= ( C
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left
thf(fact_541_add_Oinv__solve__left_H,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B ) @ C )
= A )
= ( C
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left'
thf(fact_542_add_Oinv__solve__right,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( A
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C ) ) )
= ( B
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right
thf(fact_543_add_Oinv__solve__right_H,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C ) )
= A )
= ( B
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right'
thf(fact_544_a__transpose__inv,axiom,
! [X2: set_int,Y: set_int,Z: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= Z )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ Z )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_545_local_Ominus__add,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_546_r__neg1,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_547_r__neg2,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_548_one__divides,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ A ) ) ).
% one_divides
thf(fact_549_minus__eq,axiom,
! [X2: set_int,Y: set_int] :
( ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ).
% minus_eq
thf(fact_550_l__neg,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ X2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% l_neg
thf(fact_551_minus__equality,axiom,
! [Y: set_int,X2: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 )
= Y ) ) ) ) ).
% minus_equality
thf(fact_552_r__neg,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r_neg
thf(fact_553_sum__zero__eq__neg,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( X2
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ) ) ).
% sum_zero_eq_neg
thf(fact_554_carrier__one__not__zero,axiom,
( ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
= ( ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% carrier_one_not_zero
thf(fact_555_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_556_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_557_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_558_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_559_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_560_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_561_a__inv__closed,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% a_inv_closed
thf(fact_562_local_Ominus__minus,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) )
= X2 ) ) ).
% local.minus_minus
thf(fact_563_one__closed,axiom,
member_set_int @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% one_closed
thf(fact_564_local_Ominus__zero,axiom,
( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% local.minus_zero
thf(fact_565_add_Oinv__eq__1__iff,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( X2
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_566_psubsetE,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B3 )
=> ~ ( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ( ord_le4403425263959731960et_int @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_567_psubsetE,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ( ord_le4562804192517611682et_int @ A2 @ B3 )
=> ~ ( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ord_le4317611570275147438et_int @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_568_psubset__eq,axiom,
( ord_less_set_set_int
= ( ^ [A4: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_569_psubset__eq,axiom,
( ord_le4562804192517611682et_int
= ( ^ [A4: set_set_set_int,B4: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_570_psubset__imp__subset,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B3 )
=> ( ord_le4403425263959731960et_int @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_571_psubset__imp__subset,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ( ord_le4562804192517611682et_int @ A2 @ B3 )
=> ( ord_le4317611570275147438et_int @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_572_psubset__subset__trans,axiom,
! [A2: set_set_int,B3: set_set_int,C2: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B3 )
=> ( ( ord_le4403425263959731960et_int @ B3 @ C2 )
=> ( ord_less_set_set_int @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_573_psubset__subset__trans,axiom,
! [A2: set_set_set_int,B3: set_set_set_int,C2: set_set_set_int] :
( ( ord_le4562804192517611682et_int @ A2 @ B3 )
=> ( ( ord_le4317611570275147438et_int @ B3 @ C2 )
=> ( ord_le4562804192517611682et_int @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_574_subset__not__subset__eq,axiom,
( ord_less_set_set_int
= ( ^ [A4: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ A4 @ B4 )
& ~ ( ord_le4403425263959731960et_int @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_575_subset__not__subset__eq,axiom,
( ord_le4562804192517611682et_int
= ( ^ [A4: set_set_set_int,B4: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A4 @ B4 )
& ~ ( ord_le4317611570275147438et_int @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_576_subset__psubset__trans,axiom,
! [A2: set_set_int,B3: set_set_int,C2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ( ( ord_less_set_set_int @ B3 @ C2 )
=> ( ord_less_set_set_int @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_577_subset__psubset__trans,axiom,
! [A2: set_set_set_int,B3: set_set_set_int,C2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ( ord_le4562804192517611682et_int @ B3 @ C2 )
=> ( ord_le4562804192517611682et_int @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_578_subset__iff__psubset__eq,axiom,
( ord_le4403425263959731960et_int
= ( ^ [A4: set_set_int,B4: set_set_int] :
( ( ord_less_set_set_int @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_579_subset__iff__psubset__eq,axiom,
( ord_le4317611570275147438et_int
= ( ^ [A4: set_set_set_int,B4: set_set_set_int] :
( ( ord_le4562804192517611682et_int @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_580_not__psubset__empty,axiom,
! [A2: set_set_int] :
~ ( ord_less_set_set_int @ A2 @ bot_bot_set_set_int ) ).
% not_psubset_empty
thf(fact_581_not__psubset__empty,axiom,
! [A2: set_set_set_int] :
~ ( ord_le4562804192517611682et_int @ A2 @ bot_bo2384636101374064866et_int ) ).
% not_psubset_empty
thf(fact_582_ring_Osubring__props_I5_J,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int,H: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( subfie3888952257595785920t_unit @ K @ R3 )
=> ( ( member_set_int @ H @ K )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R3 @ H ) @ K ) ) ) ) ).
% ring.subring_props(5)
thf(fact_583_ring__iso__memE_I4_J,axiom,
! [H: set_int > set_int,R3: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit] :
( ( member5205197933313416826et_int @ H @ ( ring_i7857697894474155344t_unit @ R3 @ S ) )
=> ( ( H @ ( one_se8065767436706823081t_unit @ R3 ) )
= ( one_se8065767436706823081t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_584_ring__iso__memE_I4_J,axiom,
! [H: set_int > nat,R3: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit] :
( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ R3 @ S ) )
=> ( ( H @ ( one_se8065767436706823081t_unit @ R3 ) )
= ( one_na902338870878123981t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_585_ring_Osubring__props_I3_J,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( subfie3888952257595785920t_unit @ K @ R3 )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ R3 ) @ K ) ) ) ).
% ring.subring_props(3)
thf(fact_586_ring_Oring__simprules_I3_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R3 @ X2 ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_587_ring_Oring__simprules_I3_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ R3 @ X2 ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_588_ring_Oring__simprules_I20_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( a_inv_5951419416477254493t_unit @ R3 @ ( a_inv_5951419416477254493t_unit @ R3 @ X2 ) )
= X2 ) ) ) ).
% ring.ring_simprules(20)
thf(fact_589_ring_Oring__simprules_I20_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( a_inv_2472168910397739247t_unit @ R3 @ ( a_inv_2472168910397739247t_unit @ R3 @ X2 ) )
= X2 ) ) ) ).
% ring.ring_simprules(20)
thf(fact_590_ring_Ominus__zero,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( a_inv_5951419416477254493t_unit @ R3 @ ( zero_s6269048424454532197t_unit @ R3 ) )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ).
% ring.minus_zero
thf(fact_591_cring_Ocring__simprules_I3_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R3 @ X2 ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ).
% cring.cring_simprules(3)
thf(fact_592_cring_Ocring__simprules_I3_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ R3 @ X2 ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ).
% cring.cring_simprules(3)
thf(fact_593_cring_Ocring__simprules_I21_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( a_inv_5951419416477254493t_unit @ R3 @ ( a_inv_5951419416477254493t_unit @ R3 @ X2 ) )
= X2 ) ) ) ).
% cring.cring_simprules(21)
thf(fact_594_cring_Ocring__simprules_I21_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( a_inv_2472168910397739247t_unit @ R3 @ ( a_inv_2472168910397739247t_unit @ R3 @ X2 ) )
= X2 ) ) ) ).
% cring.cring_simprules(21)
thf(fact_595_cring_Ocring__simprules_I22_J,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( a_inv_2472168910397739247t_unit @ R3 @ ( zero_n5149899317435570679t_unit @ R3 ) )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ).
% cring.cring_simprules(22)
thf(fact_596_cring_Ocring__simprules_I22_J,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( a_inv_5951419416477254493t_unit @ R3 @ ( zero_s6269048424454532197t_unit @ R3 ) )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ).
% cring.cring_simprules(22)
thf(fact_597_abelian__group_Oa__inv__closed,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ G @ X2 ) @ ( partia966996272515721803t_unit @ G ) ) ) ) ).
% abelian_group.a_inv_closed
thf(fact_598_abelian__group_Oa__inv__closed,axiom,
! [G: partia4692342223508353374t_unit,X2: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( member_nat @ ( a_inv_2472168910397739247t_unit @ G @ X2 ) @ ( partia3499330772048238685t_unit @ G ) ) ) ) ).
% abelian_group.a_inv_closed
thf(fact_599_abelian__group_Ominus__minus,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( a_inv_5951419416477254493t_unit @ G @ ( a_inv_5951419416477254493t_unit @ G @ X2 ) )
= X2 ) ) ) ).
% abelian_group.minus_minus
thf(fact_600_abelian__group_Ominus__minus,axiom,
! [G: partia4692342223508353374t_unit,X2: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( a_inv_2472168910397739247t_unit @ G @ ( a_inv_2472168910397739247t_unit @ G @ X2 ) )
= X2 ) ) ) ).
% abelian_group.minus_minus
thf(fact_601_a__minus__def,axiom,
( a_minu5974516859897376926t_unit
= ( ^ [R5: partia4934656038542163276t_unit,X4: set_int,Y6: set_int] : ( add_se5859248395121729892t_unit @ R5 @ X4 @ ( a_inv_5951419416477254493t_unit @ R5 @ Y6 ) ) ) ) ).
% a_minus_def
thf(fact_602_ring_Oring__simprules_I6_J,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ R3 ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ).
% ring.ring_simprules(6)
thf(fact_603_ring_Oring__simprules_I6_J,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ R3 ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ).
% ring.ring_simprules(6)
thf(fact_604_cring_Ocring__simprules_I6_J,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ R3 ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ).
% cring.cring_simprules(6)
thf(fact_605_cring_Ocring__simprules_I6_J,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ R3 ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ).
% cring.cring_simprules(6)
thf(fact_606_ring_Oone__divides,axiom,
! [R3: partia4934656038542163276t_unit,A: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R3 ) )
=> ( factor5186451337065598620t_unit @ R3 @ ( one_se8065767436706823081t_unit @ R3 ) @ A ) ) ) ).
% ring.one_divides
thf(fact_607_ring_Oone__divides,axiom,
! [R3: partia4692342223508353374t_unit,A: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( factor7017787129603596992t_unit @ R3 @ ( one_na902338870878123981t_unit @ R3 ) @ A ) ) ) ).
% ring.one_divides
thf(fact_608_semiring_Osemiring__simprules_I4_J,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ R3 ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_609_semiring_Osemiring__simprules_I4_J,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( member_nat @ ( one_na902338870878123981t_unit @ R3 ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_610_ring_Oup__a__inv__closed,axiom,
! [R3: partia4934656038542163276t_unit,P2: nat > set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_nat_set_int @ P2 @ ( up_set1168727741560211120t_unit @ R3 ) )
=> ( member_nat_set_int
@ ^ [I: nat] : ( a_inv_5951419416477254493t_unit @ R3 @ ( P2 @ I ) )
@ ( up_set1168727741560211120t_unit @ R3 ) ) ) ) ).
% ring.up_a_inv_closed
thf(fact_611_ring_Oline__extension__in__carrier,axiom,
! [R3: partia4692342223508353374t_unit,K: set_nat,A: nat,E: set_nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( ord_less_eq_set_nat @ K @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( ord_less_eq_set_nat @ E @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ord_less_eq_set_nat @ ( embedd838748496991043025t_unit @ R3 @ K @ A @ E ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_612_ring_Oline__extension__in__carrier,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int,A: set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ord_le4403425263959731960et_int @ ( embedd4283282269743769663t_unit @ R3 @ K @ A @ E ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ) ) ).
% ring.line_extension_in_carrier
thf(fact_613_ring_Oring__simprules_I19_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( a_inv_5951419416477254493t_unit @ R3 @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ R3 @ ( a_inv_5951419416477254493t_unit @ R3 @ X2 ) @ ( a_inv_5951419416477254493t_unit @ R3 @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_614_ring_Oring__simprules_I19_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( a_inv_2472168910397739247t_unit @ R3 @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) )
= ( add_nat_Product_unit @ R3 @ ( a_inv_2472168910397739247t_unit @ R3 @ X2 ) @ ( a_inv_2472168910397739247t_unit @ R3 @ Y ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_615_ring_Oring__simprules_I18_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( a_inv_5951419416477254493t_unit @ R3 @ X2 ) @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_616_ring_Oring__simprules_I18_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( a_inv_2472168910397739247t_unit @ R3 @ X2 ) @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_617_ring_Oring__simprules_I17_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( add_se5859248395121729892t_unit @ R3 @ ( a_inv_5951419416477254493t_unit @ R3 @ X2 ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_618_ring_Oring__simprules_I17_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X2 @ ( add_nat_Product_unit @ R3 @ ( a_inv_2472168910397739247t_unit @ R3 @ X2 ) @ Y ) )
= Y ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_619_cring_Ocring__simprules_I20_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( a_inv_5951419416477254493t_unit @ R3 @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ R3 @ ( a_inv_5951419416477254493t_unit @ R3 @ X2 ) @ ( a_inv_5951419416477254493t_unit @ R3 @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(20)
thf(fact_620_cring_Ocring__simprules_I20_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( a_inv_2472168910397739247t_unit @ R3 @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) )
= ( add_nat_Product_unit @ R3 @ ( a_inv_2472168910397739247t_unit @ R3 @ X2 ) @ ( a_inv_2472168910397739247t_unit @ R3 @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(20)
thf(fact_621_cring_Ocring__simprules_I19_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( a_inv_5951419416477254493t_unit @ R3 @ X2 ) @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) )
= Y ) ) ) ) ).
% cring.cring_simprules(19)
thf(fact_622_cring_Ocring__simprules_I19_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( a_inv_2472168910397739247t_unit @ R3 @ X2 ) @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) )
= Y ) ) ) ) ).
% cring.cring_simprules(19)
thf(fact_623_cring_Ocring__simprules_I18_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( add_se5859248395121729892t_unit @ R3 @ ( a_inv_5951419416477254493t_unit @ R3 @ X2 ) @ Y ) )
= Y ) ) ) ) ).
% cring.cring_simprules(18)
thf(fact_624_cring_Ocring__simprules_I18_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X2 @ ( add_nat_Product_unit @ R3 @ ( a_inv_2472168910397739247t_unit @ R3 @ X2 ) @ Y ) )
= Y ) ) ) ) ).
% cring.cring_simprules(18)
thf(fact_625_abelian__group_Or__neg1,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ ( a_inv_5951419416477254493t_unit @ G @ X2 ) @ ( add_se5859248395121729892t_unit @ G @ X2 @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg1
thf(fact_626_abelian__group_Or__neg1,axiom,
! [G: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( add_nat_Product_unit @ G @ ( a_inv_2472168910397739247t_unit @ G @ X2 ) @ ( add_nat_Product_unit @ G @ X2 @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg1
thf(fact_627_abelian__group_Or__neg2,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ X2 @ ( add_se5859248395121729892t_unit @ G @ ( a_inv_5951419416477254493t_unit @ G @ X2 ) @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg2
thf(fact_628_abelian__group_Or__neg2,axiom,
! [G: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( add_nat_Product_unit @ G @ X2 @ ( add_nat_Product_unit @ G @ ( a_inv_2472168910397739247t_unit @ G @ X2 ) @ Y ) )
= Y ) ) ) ) ).
% abelian_group.r_neg2
thf(fact_629_abelian__group_Ominus__add,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( a_inv_5951419416477254493t_unit @ G @ ( add_se5859248395121729892t_unit @ G @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ G @ ( a_inv_5951419416477254493t_unit @ G @ X2 ) @ ( a_inv_5951419416477254493t_unit @ G @ Y ) ) ) ) ) ) ).
% abelian_group.minus_add
thf(fact_630_abelian__group_Ominus__add,axiom,
! [G: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( a_inv_2472168910397739247t_unit @ G @ ( add_nat_Product_unit @ G @ X2 @ Y ) )
= ( add_nat_Product_unit @ G @ ( a_inv_2472168910397739247t_unit @ G @ X2 ) @ ( a_inv_2472168910397739247t_unit @ G @ Y ) ) ) ) ) ) ).
% abelian_group.minus_add
thf(fact_631_ring_Oring__simprules_I14_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( a_minu5974516859897376926t_unit @ R3 @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( a_inv_5951419416477254493t_unit @ R3 @ Y ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_632_cring_Ocring__simprules_I15_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( a_minu1751788497103602224t_unit @ R3 @ X2 @ Y )
= ( add_nat_Product_unit @ R3 @ X2 @ ( a_inv_2472168910397739247t_unit @ R3 @ Y ) ) ) ) ).
% cring.cring_simprules(15)
thf(fact_633_cring_Ocring__simprules_I15_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( a_minu5974516859897376926t_unit @ R3 @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( a_inv_5951419416477254493t_unit @ R3 @ Y ) ) ) ) ).
% cring.cring_simprules(15)
thf(fact_634_abelian__group_Ominus__eq,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( a_minu5974516859897376926t_unit @ G @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ G @ X2 @ ( a_inv_5951419416477254493t_unit @ G @ Y ) ) ) ) ).
% abelian_group.minus_eq
thf(fact_635_ring_Oring__simprules_I9_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( a_inv_5951419416477254493t_unit @ R3 @ X2 ) @ X2 )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_636_ring_Oring__simprules_I9_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( a_inv_2472168910397739247t_unit @ R3 @ X2 ) @ X2 )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_637_ring_Oring__simprules_I16_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( a_inv_5951419416477254493t_unit @ R3 @ X2 ) )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_638_ring_Oring__simprules_I16_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X2 @ ( a_inv_2472168910397739247t_unit @ R3 @ X2 ) )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_639_cring_Osum__zero__eq__neg,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y )
= ( zero_s6269048424454532197t_unit @ R3 ) )
=> ( X2
= ( a_inv_5951419416477254493t_unit @ R3 @ Y ) ) ) ) ) ) ).
% cring.sum_zero_eq_neg
thf(fact_640_cring_Osum__zero__eq__neg,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( ( add_nat_Product_unit @ R3 @ X2 @ Y )
= ( zero_n5149899317435570679t_unit @ R3 ) )
=> ( X2
= ( a_inv_2472168910397739247t_unit @ R3 @ Y ) ) ) ) ) ) ).
% cring.sum_zero_eq_neg
thf(fact_641_cring_Ocring__simprules_I9_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ ( a_inv_5951419416477254493t_unit @ R3 @ X2 ) @ X2 )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% cring.cring_simprules(9)
thf(fact_642_cring_Ocring__simprules_I9_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ ( a_inv_2472168910397739247t_unit @ R3 @ X2 ) @ X2 )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ).
% cring.cring_simprules(9)
thf(fact_643_cring_Ocring__simprules_I17_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( add_se5859248395121729892t_unit @ R3 @ X2 @ ( a_inv_5951419416477254493t_unit @ R3 @ X2 ) )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% cring.cring_simprules(17)
thf(fact_644_cring_Ocring__simprules_I17_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( add_nat_Product_unit @ R3 @ X2 @ ( a_inv_2472168910397739247t_unit @ R3 @ X2 ) )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ).
% cring.cring_simprules(17)
thf(fact_645_abelian__group_Ominus__equality,axiom,
! [G: partia4934656038542163276t_unit,Y: set_int,X2: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( ( add_se5859248395121729892t_unit @ G @ Y @ X2 )
= ( zero_s6269048424454532197t_unit @ G ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( a_inv_5951419416477254493t_unit @ G @ X2 )
= Y ) ) ) ) ) ).
% abelian_group.minus_equality
thf(fact_646_abelian__group_Ominus__equality,axiom,
! [G: partia4692342223508353374t_unit,Y: nat,X2: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( ( add_nat_Product_unit @ G @ Y @ X2 )
= ( zero_n5149899317435570679t_unit @ G ) )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( a_inv_2472168910397739247t_unit @ G @ X2 )
= Y ) ) ) ) ) ).
% abelian_group.minus_equality
thf(fact_647_abelian__group_Or__neg,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ X2 @ ( a_inv_5951419416477254493t_unit @ G @ X2 ) )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ) ).
% abelian_group.r_neg
thf(fact_648_abelian__group_Or__neg,axiom,
! [G: partia4692342223508353374t_unit,X2: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( add_nat_Product_unit @ G @ X2 @ ( a_inv_2472168910397739247t_unit @ G @ X2 ) )
= ( zero_n5149899317435570679t_unit @ G ) ) ) ) ).
% abelian_group.r_neg
thf(fact_649_abelian__group_Ol__neg,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ ( a_inv_5951419416477254493t_unit @ G @ X2 ) @ X2 )
= ( zero_s6269048424454532197t_unit @ G ) ) ) ) ).
% abelian_group.l_neg
thf(fact_650_abelian__group_Ol__neg,axiom,
! [G: partia4692342223508353374t_unit,X2: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( add_nat_Product_unit @ G @ ( a_inv_2472168910397739247t_unit @ G @ X2 ) @ X2 )
= ( zero_n5149899317435570679t_unit @ G ) ) ) ) ).
% abelian_group.l_neg
thf(fact_651_ring_Ofinite__dimension_Ocong,axiom,
embedd8246663962306818995t_unit = embedd8246663962306818995t_unit ).
% ring.finite_dimension.cong
thf(fact_652_ring_Ogenideal__one,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( genide1545711809618862555t_unit @ R3 @ ( insert_set_int @ ( one_se8065767436706823081t_unit @ R3 ) @ bot_bot_set_set_int ) )
= ( partia966996272515721803t_unit @ R3 ) ) ) ).
% ring.genideal_one
thf(fact_653_ring_Ogenideal__one,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( genide4496968333291595629t_unit @ R3 @ ( insert_nat @ ( one_na902338870878123981t_unit @ R3 ) @ bot_bot_set_nat ) )
= ( partia3499330772048238685t_unit @ R3 ) ) ) ).
% ring.genideal_one
thf(fact_654_semiring_Ocarrier__one__not__zero,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( ( partia966996272515721803t_unit @ R3 )
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ bot_bot_set_set_int ) )
= ( ( one_se8065767436706823081t_unit @ R3 )
!= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_655_semiring_Ocarrier__one__not__zero,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( ( partia3499330772048238685t_unit @ R3 )
!= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ bot_bot_set_nat ) )
= ( ( one_na902338870878123981t_unit @ R3 )
!= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_656_semiring_Ocarrier__one__zero,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( ( partia966996272515721803t_unit @ R3 )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ bot_bot_set_set_int ) )
= ( ( one_se8065767436706823081t_unit @ R3 )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_657_semiring_Ocarrier__one__zero,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( ( partia3499330772048238685t_unit @ R3 )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ bot_bot_set_nat ) )
= ( ( one_na902338870878123981t_unit @ R3 )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_658_semiring_Oone__zeroI,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( ( partia966996272515721803t_unit @ R3 )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ bot_bot_set_set_int ) )
=> ( ( one_se8065767436706823081t_unit @ R3 )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% semiring.one_zeroI
thf(fact_659_semiring_Oone__zeroI,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( ( partia3499330772048238685t_unit @ R3 )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ bot_bot_set_nat ) )
=> ( ( one_na902338870878123981t_unit @ R3 )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ).
% semiring.one_zeroI
thf(fact_660_semiring_Oone__zeroD,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( ( one_se8065767436706823081t_unit @ R3 )
= ( zero_s6269048424454532197t_unit @ R3 ) )
=> ( ( partia966996272515721803t_unit @ R3 )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ bot_bot_set_set_int ) ) ) ) ).
% semiring.one_zeroD
thf(fact_661_semiring_Oone__zeroD,axiom,
! [R3: partia4692342223508353374t_unit] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( ( one_na902338870878123981t_unit @ R3 )
= ( zero_n5149899317435570679t_unit @ R3 ) )
=> ( ( partia3499330772048238685t_unit @ R3 )
= ( insert_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ bot_bot_set_nat ) ) ) ) ).
% semiring.one_zeroD
thf(fact_662_zfact__iso__inv__is__ring__iso,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( member_set_int_nat @ ( zfact_iso_inv @ N ) @ ( ring_i6162119212153773794t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N ) ) @ ( mod_ring @ N ) ) ) ) ).
% zfact_iso_inv_is_ring_iso
thf(fact_663_ring_Osubring__props_I2_J,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( subfie3888952257595785920t_unit @ K @ R3 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ K ) ) ) ).
% ring.subring_props(2)
thf(fact_664_ring_Osubring__props_I7_J,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int,H1: set_int,H2: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( subfie3888952257595785920t_unit @ K @ R3 )
=> ( ( member_set_int @ H1 @ K )
=> ( ( member_set_int @ H2 @ K )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ R3 @ H1 @ H2 ) @ K ) ) ) ) ) ).
% ring.subring_props(7)
thf(fact_665_ring_Osubring__props_I4_J,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( subfie3888952257595785920t_unit @ K @ R3 )
=> ( K != bot_bot_set_set_int ) ) ) ).
% ring.subring_props(4)
thf(fact_666_ring_Ozero__divides,axiom,
! [R3: partia4934656038542163276t_unit,A: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( factor5186451337065598620t_unit @ R3 @ ( zero_s6269048424454532197t_unit @ R3 ) @ A )
= ( A
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% ring.zero_divides
thf(fact_667_ring_Otelescopic__base__dim_I1_J,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int,F3: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( subfie3888952257595785920t_unit @ K @ R3 )
=> ( ( subfie3888952257595785920t_unit @ F3 @ R3 )
=> ( ( embedd8246663962306818995t_unit @ R3 @ K @ F3 )
=> ( ( embedd8246663962306818995t_unit @ R3 @ F3 @ E )
=> ( embedd8246663962306818995t_unit @ R3 @ K @ E ) ) ) ) ) ) ).
% ring.telescopic_base_dim(1)
thf(fact_668_ring_Osubring__props_I1_J,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( subfie3888952257595785920t_unit @ K @ R3 )
=> ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ).
% ring.subring_props(1)
thf(fact_669_ring_Osubring__props_I1_J,axiom,
! [R3: partia4692342223508353374t_unit,K: set_nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( subfie4892355163478727762t_unit @ K @ R3 )
=> ( ord_less_eq_set_nat @ K @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ).
% ring.subring_props(1)
thf(fact_670_ring_Odivides__zero,axiom,
! [R3: partia4934656038542163276t_unit,A: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R3 ) )
=> ( factor5186451337065598620t_unit @ R3 @ A @ ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% ring.divides_zero
thf(fact_671_ring_Odivides__zero,axiom,
! [R3: partia4692342223508353374t_unit,A: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( factor7017787129603596992t_unit @ R3 @ A @ ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ).
% ring.divides_zero
thf(fact_672_ring_Ocarrier__is__subalgebra,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ R3 ) )
=> ( embedd2743979684206749024t_unit @ K @ ( partia966996272515721803t_unit @ R3 ) @ R3 ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_673_ring_Ocarrier__is__subalgebra,axiom,
! [R3: partia4692342223508353374t_unit,K: set_nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( ord_less_eq_set_nat @ K @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( embedd2609395410403458802t_unit @ K @ ( partia3499330772048238685t_unit @ R3 ) @ R3 ) ) ) ).
% ring.carrier_is_subalgebra
thf(fact_674_ring_Osubalgebra__in__carrier,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int,V3: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( embedd2743979684206749024t_unit @ K @ V3 @ R3 )
=> ( ord_le4403425263959731960et_int @ V3 @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_675_ring_Osubalgebra__in__carrier,axiom,
! [R3: partia4692342223508353374t_unit,K: set_nat,V3: set_nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( embedd2609395410403458802t_unit @ K @ V3 @ R3 )
=> ( ord_less_eq_set_nat @ V3 @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ).
% ring.subalgebra_in_carrier
thf(fact_676_ring_Osum__space__dim_I1_J,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int,E: set_set_int,F3: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( subfie3888952257595785920t_unit @ K @ R3 )
=> ( ( embedd8246663962306818995t_unit @ R3 @ K @ E )
=> ( ( embedd8246663962306818995t_unit @ R3 @ K @ F3 )
=> ( embedd8246663962306818995t_unit @ R3 @ K @ ( set_ad273131178244904872t_unit @ R3 @ E @ F3 ) ) ) ) ) ) ).
% ring.sum_space_dim(1)
thf(fact_677_cring_Oassociated__iff__same__ideal,axiom,
! [R3: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( associ3816808824722140549t_unit @ R3 @ A @ B )
= ( ( cgenid8502489213727343375t_unit @ R3 @ A )
= ( cgenid8502489213727343375t_unit @ R3 @ B ) ) ) ) ) ) ).
% cring.associated_iff_same_ideal
thf(fact_678_cring_Oassociated__iff__same__ideal,axiom,
! [R3: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( associ4357985397585971625t_unit @ R3 @ A @ B )
= ( ( cgenid8693976350862946099t_unit @ R3 @ A )
= ( cgenid8693976350862946099t_unit @ R3 @ B ) ) ) ) ) ) ).
% cring.associated_iff_same_ideal
thf(fact_679_ring_Ofinite__dimension__imp__subalgebra,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( subfie3888952257595785920t_unit @ K @ R3 )
=> ( ( embedd8246663962306818995t_unit @ R3 @ K @ E )
=> ( embedd2743979684206749024t_unit @ K @ E @ R3 ) ) ) ) ).
% ring.finite_dimension_imp_subalgebra
thf(fact_680_up__one__closed,axiom,
( member_nat_set_int
@ ^ [N4: nat] : ( if_set_int @ ( N4 = zero_zero_nat ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
@ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% up_one_closed
thf(fact_681_line__extension__smult__closed,axiom,
! [K: set_set_int,E: set_set_int,A: set_int,K2: set_int,U: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ! [K3: set_int,V2: set_int] :
( ( member_set_int @ K3 @ K )
=> ( ( member_set_int @ V2 @ E )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K3 @ V2 ) @ E ) ) )
=> ( ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ K2 @ K )
=> ( ( member_set_int @ U @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ A @ E ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ U ) @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ A @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_682_abelian__group_Oa__lcos__m__assoc,axiom,
! [G: partia4934656038542163276t_unit,M2: set_set_int,G2: set_int,H: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ G2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ H @ ( partia966996272515721803t_unit @ G ) )
=> ( ( a_l_co3504123944629134560t_unit @ G @ G2 @ ( a_l_co3504123944629134560t_unit @ G @ H @ M2 ) )
= ( a_l_co3504123944629134560t_unit @ G @ ( add_se5859248395121729892t_unit @ G @ G2 @ H ) @ M2 ) ) ) ) ) ) ).
% abelian_group.a_lcos_m_assoc
thf(fact_683_abelian__group_Oa__lcos__m__assoc,axiom,
! [G: partia4692342223508353374t_unit,M2: set_nat,G2: nat,H: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ G2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ H @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( a_l_co3340896127261556338t_unit @ G @ G2 @ ( a_l_co3340896127261556338t_unit @ G @ H @ M2 ) )
= ( a_l_co3340896127261556338t_unit @ G @ ( add_nat_Product_unit @ G @ G2 @ H ) @ M2 ) ) ) ) ) ) ).
% abelian_group.a_lcos_m_assoc
thf(fact_684_abelian__group_Oa__lcos__mult__one,axiom,
! [G: partia4934656038542163276t_unit,M2: set_set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( a_l_co3504123944629134560t_unit @ G @ ( zero_s6269048424454532197t_unit @ G ) @ M2 )
= M2 ) ) ) ).
% abelian_group.a_lcos_mult_one
thf(fact_685_abelian__group_Oa__lcos__mult__one,axiom,
! [G: partia4692342223508353374t_unit,M2: set_nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( a_l_co3340896127261556338t_unit @ G @ ( zero_n5149899317435570679t_unit @ G ) @ M2 )
= M2 ) ) ) ).
% abelian_group.a_lcos_mult_one
thf(fact_686_subcring__iff,axiom,
! [H3: set_set_int] :
( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( subcri1024317279029940167t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( cring_3079150759069666002t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : H3
@ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% subcring_iff
thf(fact_687_a__rcos__assoc__lcos,axiom,
! [H3: set_set_int,K: set_set_int,X2: set_int] :
( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H3 @ X2 ) @ K )
= ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H3 @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ K ) ) ) ) ) ) ).
% a_rcos_assoc_lcos
thf(fact_688_add_Oint__pow__inv,axiom,
! [X2: set_int,I2: int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ X2 ) ) ) ) ).
% add.int_pow_inv
thf(fact_689_n__ge__0,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% n_ge_0
thf(fact_690_m__lcomm,axiom,
! [X2: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_691_m__comm,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 ) ) ) ) ).
% m_comm
thf(fact_692_m__assoc,axiom,
! [X2: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_693_subring__props_I6_J,axiom,
! [K: set_set_int,H1: set_int,H2: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ H1 @ K )
=> ( ( member_set_int @ H2 @ K )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H1 @ H2 ) @ K ) ) ) ) ).
% subring_props(6)
thf(fact_694_r__distr,axiom,
! [X2: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ X2 ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_695_l__distr,axiom,
! [X2: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Z ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_696_add_Oint__pow__mult__distrib,axiom,
! [X2: set_int,Y: set_int,I2: int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ X2 ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ Y ) ) ) ) ) ) ).
% add.int_pow_mult_distrib
thf(fact_697_add_Oint__pow__distrib,axiom,
! [X2: set_int,Y: set_int,I2: int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ X2 ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ Y ) ) ) ) ) ).
% add.int_pow_distrib
thf(fact_698_r__minus,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) ) ) ) ) ).
% r_minus
thf(fact_699_l__minus,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ Y )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) ) ) ) ) ).
% l_minus
thf(fact_700_one__unique,axiom,
! [U: set_int] :
( ( member_set_int @ U @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ U @ X )
= X ) )
=> ( U
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% one_unique
thf(fact_701_inv__unique,axiom,
! [Y: set_int,X2: set_int,Y2: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y2 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% inv_unique
thf(fact_702_mult__cong__r,axiom,
! [B: set_int,B6: set_int,A: set_int] :
( ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ B6 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B6 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B6 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_703_mult__cong__l,axiom,
! [A: set_int,A6: set_int,B: set_int] :
( ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ A6 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A6 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A6 @ B ) ) ) ) ) ) ).
% mult_cong_l
thf(fact_704_divides__prod__r,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ C ) ) ) ) ) ).
% divides_prod_r
thf(fact_705_divides__prod__l,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C @ B ) ) ) ) ) ) ).
% divides_prod_l
thf(fact_706_local_Odivides__mult,axiom,
! [A: set_int,C: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C @ A ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C @ B ) ) ) ) ) ).
% local.divides_mult
thf(fact_707_add__pow__rdistr__int,axiom,
! [A: set_int,B: set_int,K2: int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ B ) )
= ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ) ) ) ).
% add_pow_rdistr_int
thf(fact_708_add__pow__ldistr__int,axiom,
! [A: set_int,B: set_int,K2: int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ A ) @ B )
= ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ) ) ) ).
% add_pow_ldistr_int
thf(fact_709_a__r__coset__subset__G,axiom,
! [H3: set_set_int,X2: set_int] :
( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H3 @ X2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% a_r_coset_subset_G
thf(fact_710_line__extension__mem__iff,axiom,
! [U: set_int,K: set_set_int,A: set_int,E: set_set_int] :
( ( member_set_int @ U @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ A @ E ) )
= ( ? [X4: set_int] :
( ( member_set_int @ X4 @ K )
& ? [Y6: set_int] :
( ( member_set_int @ Y6 @ E )
& ( U
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X4 @ A ) @ Y6 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_711_properfactor__prod__r,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ C ) ) ) ) ) ) ).
% properfactor_prod_r
thf(fact_712_properfactor__prod__l,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C @ B ) ) ) ) ) ) ).
% properfactor_prod_l
thf(fact_713_up__smult__closed,axiom,
! [A: set_int,P2: nat > set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_nat_set_int @ P2 @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_nat_set_int
@ ^ [I: nat] : ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ ( P2 @ I ) )
@ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% up_smult_closed
thf(fact_714_a__rcosI,axiom,
! [H: set_int,H3: set_set_int,X2: set_int] :
( ( member_set_int @ H @ H3 )
=> ( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H @ X2 ) @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H3 @ X2 ) ) ) ) ) ).
% a_rcosI
thf(fact_715_a__coset__add__assoc,axiom,
! [M2: set_set_int,G2: set_int,H: set_int] :
( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ G2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M2 @ G2 ) @ H )
= ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M2 @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ G2 @ H ) ) ) ) ) ) ).
% a_coset_add_assoc
thf(fact_716_a__setmult__rcos__assoc,axiom,
! [H3: set_set_int,K: set_set_int,X2: set_int] :
( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H3 @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ X2 ) )
= ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H3 @ K ) @ X2 ) ) ) ) ) ).
% a_setmult_rcos_assoc
thf(fact_717_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_718_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_719_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_720_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_721_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_722_a__coset__add__inv2,axiom,
! [M2: set_set_int,X2: set_int,Y: set_int] :
( ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M2 @ X2 )
= ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M2 @ Y ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M2 @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) )
= M2 ) ) ) ) ) ).
% a_coset_add_inv2
thf(fact_723_a__coset__add__inv1,axiom,
! [M2: set_set_int,X2: set_int,Y: set_int] :
( ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M2 @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) )
= M2 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M2 @ X2 )
= ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M2 @ Y ) ) ) ) ) ) ).
% a_coset_add_inv1
thf(fact_724_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_725_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_726_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_727_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_728_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_729_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_730_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_731_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_732_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_733_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_734_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_735_m__closed,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% m_closed
thf(fact_736_add_Oint__pow__1,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ one_one_int @ X2 )
= X2 ) ) ).
% add.int_pow_1
thf(fact_737_add_Oint__pow__closed,axiom,
! [X2: set_int,I2: int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ X2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% add.int_pow_closed
thf(fact_738_add_Oint__pow__one,axiom,
! [Z: int] :
( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% add.int_pow_one
thf(fact_739_r__null,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r_null
thf(fact_740_l__null,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% l_null
thf(fact_741_r__one,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= X2 ) ) ).
% r_one
thf(fact_742_l__one,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X2 )
= X2 ) ) ).
% l_one
thf(fact_743_divides__mult__rI,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ C ) ) ) ) ) ) ).
% divides_mult_rI
thf(fact_744_divides__mult__lI,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C @ A ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C @ B ) ) ) ) ) ).
% divides_mult_lI
thf(fact_745_a__coset__add__zero,axiom,
! [M2: set_set_int] :
( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= M2 ) ) ).
% a_coset_add_zero
thf(fact_746_psubsetD,axiom,
! [A2: set_set_int,B3: set_set_int,C: set_int] :
( ( ord_less_set_set_int @ A2 @ B3 )
=> ( ( member_set_int @ C @ A2 )
=> ( member_set_int @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_747_psubsetD,axiom,
! [A2: set_nat_set_int,B3: set_nat_set_int,C: nat > set_int] :
( ( ord_le2931775347370382171et_int @ A2 @ B3 )
=> ( ( member_nat_set_int @ C @ A2 )
=> ( member_nat_set_int @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_748_psubsetD,axiom,
! [A2: set_nat,B3: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_749_psubsetD,axiom,
! [A2: set_set_int_nat,B3: set_set_int_nat,C: set_int > nat] :
( ( ord_le4941564535344212059nt_nat @ A2 @ B3 )
=> ( ( member_set_int_nat @ C @ A2 )
=> ( member_set_int_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_750_psubsetD,axiom,
! [A2: set_set_set_int,B3: set_set_set_int,C: set_set_int] :
( ( ord_le4562804192517611682et_int @ A2 @ B3 )
=> ( ( member_set_set_int @ C @ A2 )
=> ( member_set_set_int @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_751_less__set__def,axiom,
( ord_less_set_set_int
= ( ^ [A4: set_set_int,B4: set_set_int] :
( ord_less_set_int_o
@ ^ [X4: set_int] : ( member_set_int @ X4 @ A4 )
@ ^ [X4: set_int] : ( member_set_int @ X4 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_752_less__set__def,axiom,
( ord_le2931775347370382171et_int
= ( ^ [A4: set_nat_set_int,B4: set_nat_set_int] :
( ord_le2435888750224449770_int_o
@ ^ [X4: nat > set_int] : ( member_nat_set_int @ X4 @ A4 )
@ ^ [X4: nat > set_int] : ( member_nat_set_int @ X4 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_753_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_754_less__set__def,axiom,
( ord_le4941564535344212059nt_nat
= ( ^ [A4: set_set_int_nat,B4: set_set_int_nat] :
( ord_le4908975622582176746_nat_o
@ ^ [X4: set_int > nat] : ( member_set_int_nat @ X4 @ A4 )
@ ^ [X4: set_int > nat] : ( member_set_int_nat @ X4 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_755_less__set__def,axiom,
( ord_le4562804192517611682et_int
= ( ^ [A4: set_set_set_int,B4: set_set_set_int] :
( ord_le538297080920705635_int_o
@ ^ [X4: set_set_int] : ( member_set_set_int @ X4 @ A4 )
@ ^ [X4: set_set_int] : ( member_set_set_int @ X4 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_756_ring_Oadd__pow__rdistr__int,axiom,
! [R3: partia4934656038542163276t_unit,A: set_int,B: set_int,K2: int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ A @ ( add_po7581009264371422883it_int @ R3 @ K2 @ B ) )
= ( add_po7581009264371422883it_int @ R3 @ K2 @ ( mult_s3864001451298473021t_unit @ R3 @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_rdistr_int
thf(fact_757_ring_Oadd__pow__rdistr__int,axiom,
! [R3: partia4692342223508353374t_unit,A: nat,B: nat,K2: int] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ A @ ( add_po2420080144553951285it_int @ R3 @ K2 @ B ) )
= ( add_po2420080144553951285it_int @ R3 @ K2 @ ( mult_n6028127365542633569t_unit @ R3 @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_rdistr_int
thf(fact_758_ring_Oadd__pow__ldistr__int,axiom,
! [R3: partia4934656038542163276t_unit,A: set_int,B: set_int,K2: int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( add_po7581009264371422883it_int @ R3 @ K2 @ A ) @ B )
= ( add_po7581009264371422883it_int @ R3 @ K2 @ ( mult_s3864001451298473021t_unit @ R3 @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_ldistr_int
thf(fact_759_ring_Oadd__pow__ldistr__int,axiom,
! [R3: partia4692342223508353374t_unit,A: nat,B: nat,K2: int] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( add_po2420080144553951285it_int @ R3 @ K2 @ A ) @ B )
= ( add_po2420080144553951285it_int @ R3 @ K2 @ ( mult_n6028127365542633569t_unit @ R3 @ A @ B ) ) ) ) ) ) ).
% ring.add_pow_ldistr_int
thf(fact_760_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_761_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_762_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_763_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_764_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_765_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_766_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_767_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_768_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_769_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_770_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_771_subalgebra_Osmult__closed,axiom,
! [K: set_set_int,V3: set_set_int,R3: partia4934656038542163276t_unit,K2: set_int,V4: set_int] :
( ( embedd2743979684206749024t_unit @ K @ V3 @ R3 )
=> ( ( member_set_int @ K2 @ K )
=> ( ( member_set_int @ V4 @ V3 )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R3 @ K2 @ V4 ) @ V3 ) ) ) ) ).
% subalgebra.smult_closed
thf(fact_772_ring_Oring__simprules_I11_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ R3 @ X2 @ ( mult_s3864001451298473021t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_773_ring_Oring__simprules_I11_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ X2 @ Y ) @ Z )
= ( mult_n6028127365542633569t_unit @ R3 @ X2 @ ( mult_n6028127365542633569t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_774_ring_Oring__simprules_I5_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_775_ring_Oring__simprules_I5_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R3 @ X2 @ Y ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_776_abelian__monoid_Oa__r__coset__subset__G,axiom,
! [G: partia4934656038542163276t_unit,H3: set_set_int,X2: set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ord_le4403425263959731960et_int @ ( a_r_co692709266861932262t_unit @ G @ H3 @ X2 ) @ ( partia966996272515721803t_unit @ G ) ) ) ) ) ).
% abelian_monoid.a_r_coset_subset_G
thf(fact_777_abelian__monoid_Oa__r__coset__subset__G,axiom,
! [G: partia4692342223508353374t_unit,H3: set_nat,X2: nat] :
( ( abelia362511065248671243t_unit @ G )
=> ( ( ord_less_eq_set_nat @ H3 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ord_less_eq_set_nat @ ( a_r_co9089071853028257912t_unit @ G @ H3 @ X2 ) @ ( partia3499330772048238685t_unit @ G ) ) ) ) ) ).
% abelian_monoid.a_r_coset_subset_G
thf(fact_778_cring_Ocring__simprules_I24_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ X2 @ ( mult_s3864001451298473021t_unit @ R3 @ Y @ Z ) )
= ( mult_s3864001451298473021t_unit @ R3 @ Y @ ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(24)
thf(fact_779_cring_Ocring__simprules_I24_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ X2 @ ( mult_n6028127365542633569t_unit @ R3 @ Y @ Z ) )
= ( mult_n6028127365542633569t_unit @ R3 @ Y @ ( mult_n6028127365542633569t_unit @ R3 @ X2 @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(24)
thf(fact_780_cring_Ocring__simprules_I14_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Y )
= ( mult_s3864001451298473021t_unit @ R3 @ Y @ X2 ) ) ) ) ) ).
% cring.cring_simprules(14)
thf(fact_781_cring_Ocring__simprules_I14_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ X2 @ Y )
= ( mult_n6028127365542633569t_unit @ R3 @ Y @ X2 ) ) ) ) ) ).
% cring.cring_simprules(14)
thf(fact_782_cring_Ocring__simprules_I11_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ R3 @ X2 @ ( mult_s3864001451298473021t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(11)
thf(fact_783_cring_Ocring__simprules_I11_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ X2 @ Y ) @ Z )
= ( mult_n6028127365542633569t_unit @ R3 @ X2 @ ( mult_n6028127365542633569t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(11)
thf(fact_784_cring_Ocring__simprules_I5_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ) ).
% cring.cring_simprules(5)
thf(fact_785_cring_Ocring__simprules_I5_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R3 @ X2 @ Y ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ) ).
% cring.cring_simprules(5)
thf(fact_786_ring_Osubring__props_I6_J,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int,H1: set_int,H2: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( subfie3888952257595785920t_unit @ K @ R3 )
=> ( ( member_set_int @ H1 @ K )
=> ( ( member_set_int @ H2 @ K )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R3 @ H1 @ H2 ) @ K ) ) ) ) ) ).
% ring.subring_props(6)
thf(fact_787_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_788_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_789_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_790_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_791_ring__iso__memE_I2_J,axiom,
! [H: set_int > set_int,R3: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( member5205197933313416826et_int @ H @ ( ring_i7857697894474155344t_unit @ R3 @ S ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( H @ ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Y ) )
= ( mult_s3864001451298473021t_unit @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_792_ring__iso__memE_I2_J,axiom,
! [H: set_int > nat,R3: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,X2: set_int,Y: set_int] :
( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ R3 @ S ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( H @ ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Y ) )
= ( mult_n6028127365542633569t_unit @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_793_ring__iso__memE_I2_J,axiom,
! [H: nat > set_int,R3: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,X2: nat,Y: nat] :
( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R3 @ S ) )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( H @ ( mult_n6028127365542633569t_unit @ R3 @ X2 @ Y ) )
= ( mult_s3864001451298473021t_unit @ S @ ( H @ X2 ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_794_semiring_Osemiring__simprules_I8_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ R3 @ X2 @ ( mult_s3864001451298473021t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_795_semiring_Osemiring__simprules_I8_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ X2 @ Y ) @ Z )
= ( mult_n6028127365542633569t_unit @ R3 @ X2 @ ( mult_n6028127365542633569t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_796_semiring_Osemiring__simprules_I3_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Y ) @ ( partia966996272515721803t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_797_semiring_Osemiring__simprules_I3_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R3 @ X2 @ Y ) @ ( partia3499330772048238685t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_798_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_799_abelian__group_Oa__coset__add__zero,axiom,
! [G: partia4934656038542163276t_unit,M2: set_set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( a_r_co692709266861932262t_unit @ G @ M2 @ ( zero_s6269048424454532197t_unit @ G ) )
= M2 ) ) ) ).
% abelian_group.a_coset_add_zero
thf(fact_800_abelian__group_Oa__coset__add__zero,axiom,
! [G: partia4692342223508353374t_unit,M2: set_nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( a_r_co9089071853028257912t_unit @ G @ M2 @ ( zero_n5149899317435570679t_unit @ G ) )
= M2 ) ) ) ).
% abelian_group.a_coset_add_zero
thf(fact_801_abelian__group_Oa__coset__add__assoc,axiom,
! [G: partia4934656038542163276t_unit,M2: set_set_int,G2: set_int,H: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ G2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ H @ ( partia966996272515721803t_unit @ G ) )
=> ( ( a_r_co692709266861932262t_unit @ G @ ( a_r_co692709266861932262t_unit @ G @ M2 @ G2 ) @ H )
= ( a_r_co692709266861932262t_unit @ G @ M2 @ ( add_se5859248395121729892t_unit @ G @ G2 @ H ) ) ) ) ) ) ) ).
% abelian_group.a_coset_add_assoc
thf(fact_802_abelian__group_Oa__coset__add__assoc,axiom,
! [G: partia4692342223508353374t_unit,M2: set_nat,G2: nat,H: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ G2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ H @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( a_r_co9089071853028257912t_unit @ G @ ( a_r_co9089071853028257912t_unit @ G @ M2 @ G2 ) @ H )
= ( a_r_co9089071853028257912t_unit @ G @ M2 @ ( add_nat_Product_unit @ G @ G2 @ H ) ) ) ) ) ) ) ).
% abelian_group.a_coset_add_assoc
thf(fact_803_abelian__group_Oa__rcosI,axiom,
! [G: partia4934656038542163276t_unit,H: set_int,H3: set_set_int,X2: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( member_set_int @ H @ H3 )
=> ( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ G @ H @ X2 ) @ ( a_r_co692709266861932262t_unit @ G @ H3 @ X2 ) ) ) ) ) ) ).
% abelian_group.a_rcosI
thf(fact_804_abelian__group_Oa__rcosI,axiom,
! [G: partia4692342223508353374t_unit,H: nat,H3: set_nat,X2: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( member_nat @ H @ H3 )
=> ( ( ord_less_eq_set_nat @ H3 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( member_nat @ ( add_nat_Product_unit @ G @ H @ X2 ) @ ( a_r_co9089071853028257912t_unit @ G @ H3 @ X2 ) ) ) ) ) ) ).
% abelian_group.a_rcosI
thf(fact_805_abelian__group_Oa__setmult__rcos__assoc,axiom,
! [G: partia4934656038542163276t_unit,H3: set_set_int,K: set_set_int,X2: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( set_ad273131178244904872t_unit @ G @ H3 @ ( a_r_co692709266861932262t_unit @ G @ K @ X2 ) )
= ( a_r_co692709266861932262t_unit @ G @ ( set_ad273131178244904872t_unit @ G @ H3 @ K ) @ X2 ) ) ) ) ) ) ).
% abelian_group.a_setmult_rcos_assoc
thf(fact_806_abelian__group_Oa__setmult__rcos__assoc,axiom,
! [G: partia4692342223508353374t_unit,H3: set_nat,K: set_nat,X2: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( ord_less_eq_set_nat @ H3 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( ord_less_eq_set_nat @ K @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( set_ad320919470248169786t_unit @ G @ H3 @ ( a_r_co9089071853028257912t_unit @ G @ K @ X2 ) )
= ( a_r_co9089071853028257912t_unit @ G @ ( set_ad320919470248169786t_unit @ G @ H3 @ K ) @ X2 ) ) ) ) ) ) ).
% abelian_group.a_setmult_rcos_assoc
thf(fact_807_ring_Oring__simprules_I25_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ X2 @ ( zero_s6269048424454532197t_unit @ R3 ) )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_808_ring_Oring__simprules_I25_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ X2 @ ( zero_n5149899317435570679t_unit @ R3 ) )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_809_ring_Oring__simprules_I24_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( zero_s6269048424454532197t_unit @ R3 ) @ X2 )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_810_ring_Oring__simprules_I24_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( zero_n5149899317435570679t_unit @ R3 ) @ X2 )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_811_ring_Oring__simprules_I13_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Z ) @ ( mult_s3864001451298473021t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_812_ring_Oring__simprules_I13_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_nat_Product_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ X2 @ Z ) @ ( mult_n6028127365542633569t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_813_ring_Oring__simprules_I23_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ Z @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ Z @ X2 ) @ ( mult_s3864001451298473021t_unit @ R3 @ Z @ Y ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_814_ring_Oring__simprules_I23_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ Z @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) )
= ( add_nat_Product_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ Z @ X2 ) @ ( mult_n6028127365542633569t_unit @ R3 @ Z @ Y ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_815_ring_Oring__simprules_I12_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( one_se8065767436706823081t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% ring.ring_simprules(12)
thf(fact_816_ring_Oring__simprules_I12_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( one_na902338870878123981t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% ring.ring_simprules(12)
thf(fact_817_cring_Ocring__simprules_I27_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ X2 @ ( zero_s6269048424454532197t_unit @ R3 ) )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% cring.cring_simprules(27)
thf(fact_818_cring_Ocring__simprules_I27_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ X2 @ ( zero_n5149899317435570679t_unit @ R3 ) )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ).
% cring.cring_simprules(27)
thf(fact_819_cring_Ocring__simprules_I26_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( zero_s6269048424454532197t_unit @ R3 ) @ X2 )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% cring.cring_simprules(26)
thf(fact_820_cring_Ocring__simprules_I26_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( zero_n5149899317435570679t_unit @ R3 ) @ X2 )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ).
% cring.cring_simprules(26)
thf(fact_821_ring_Or__minus,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ X2 @ ( a_inv_5951419416477254493t_unit @ R3 @ Y ) )
= ( a_inv_5951419416477254493t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Y ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_822_ring_Or__minus,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ X2 @ ( a_inv_2472168910397739247t_unit @ R3 @ Y ) )
= ( a_inv_2472168910397739247t_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ X2 @ Y ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_823_ring_Ol__minus,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( a_inv_5951419416477254493t_unit @ R3 @ X2 ) @ Y )
= ( a_inv_5951419416477254493t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Y ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_824_ring_Ol__minus,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( a_inv_2472168910397739247t_unit @ R3 @ X2 ) @ Y )
= ( a_inv_2472168910397739247t_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ X2 @ Y ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_825_cring_Ocring__simprules_I13_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Z ) @ ( mult_s3864001451298473021t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(13)
thf(fact_826_cring_Ocring__simprules_I13_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_nat_Product_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ X2 @ Z ) @ ( mult_n6028127365542633569t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(13)
thf(fact_827_cring_Ocring__simprules_I25_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ Z @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ Z @ X2 ) @ ( mult_s3864001451298473021t_unit @ R3 @ Z @ Y ) ) ) ) ) ) ) ).
% cring.cring_simprules(25)
thf(fact_828_cring_Ocring__simprules_I25_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ Z @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) )
= ( add_nat_Product_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ Z @ X2 ) @ ( mult_n6028127365542633569t_unit @ R3 @ Z @ Y ) ) ) ) ) ) ) ).
% cring.cring_simprules(25)
thf(fact_829_cring_Ocring__simprules_I12_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( one_se8065767436706823081t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% cring.cring_simprules(12)
thf(fact_830_cring_Ocring__simprules_I12_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( one_na902338870878123981t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% cring.cring_simprules(12)
thf(fact_831_cring_Ocring__simprules_I28_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( a_inv_5951419416477254493t_unit @ R3 @ X2 ) @ Y )
= ( a_inv_5951419416477254493t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(28)
thf(fact_832_cring_Ocring__simprules_I28_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( a_inv_2472168910397739247t_unit @ R3 @ X2 ) @ Y )
= ( a_inv_2472168910397739247t_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ X2 @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(28)
thf(fact_833_cring_Ocring__simprules_I29_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ X2 @ ( a_inv_5951419416477254493t_unit @ R3 @ Y ) )
= ( a_inv_5951419416477254493t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(29)
thf(fact_834_cring_Ocring__simprules_I29_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ X2 @ ( a_inv_2472168910397739247t_unit @ R3 @ Y ) )
= ( a_inv_2472168910397739247t_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ X2 @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(29)
thf(fact_835_ring_Odivides__mult,axiom,
! [R3: partia4934656038542163276t_unit,A: set_int,C: set_int,B: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( factor5186451337065598620t_unit @ R3 @ A @ B )
=> ( factor5186451337065598620t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ C @ A ) @ ( mult_s3864001451298473021t_unit @ R3 @ C @ B ) ) ) ) ) ) ).
% ring.divides_mult
thf(fact_836_ring_Odivides__mult,axiom,
! [R3: partia4692342223508353374t_unit,A: nat,C: nat,B: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( factor7017787129603596992t_unit @ R3 @ A @ B )
=> ( factor7017787129603596992t_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ C @ A ) @ ( mult_n6028127365542633569t_unit @ R3 @ C @ B ) ) ) ) ) ) ).
% ring.divides_mult
thf(fact_837_ring_Oline__extension__mem__iff,axiom,
! [R3: partia4934656038542163276t_unit,U: set_int,K: set_set_int,A: set_int,E: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ U @ ( embedd4283282269743769663t_unit @ R3 @ K @ A @ E ) )
= ( ? [X4: set_int] :
( ( member_set_int @ X4 @ K )
& ? [Y6: set_int] :
( ( member_set_int @ Y6 @ E )
& ( U
= ( add_se5859248395121729892t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ X4 @ A ) @ Y6 ) ) ) ) ) ) ) ).
% ring.line_extension_mem_iff
thf(fact_838_semiring_Ol__null,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( zero_s6269048424454532197t_unit @ R3 ) @ X2 )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% semiring.l_null
thf(fact_839_semiring_Ol__null,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( zero_n5149899317435570679t_unit @ R3 ) @ X2 )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ).
% semiring.l_null
thf(fact_840_semiring_Or__null,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ X2 @ ( zero_s6269048424454532197t_unit @ R3 ) )
= ( zero_s6269048424454532197t_unit @ R3 ) ) ) ) ).
% semiring.r_null
thf(fact_841_semiring_Or__null,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ X2 @ ( zero_n5149899317435570679t_unit @ R3 ) )
= ( zero_n5149899317435570679t_unit @ R3 ) ) ) ) ).
% semiring.r_null
thf(fact_842_semiring_Or__distr,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ Z @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) )
= ( add_se5859248395121729892t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ Z @ X2 ) @ ( mult_s3864001451298473021t_unit @ R3 @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_843_semiring_Or__distr,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ Z @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) )
= ( add_nat_Product_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ Z @ X2 ) @ ( mult_n6028127365542633569t_unit @ R3 @ Z @ Y ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_844_semiring_Ol__distr,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( add_se5859248395121729892t_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ R3 @ ( mult_s3864001451298473021t_unit @ R3 @ X2 @ Z ) @ ( mult_s3864001451298473021t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_845_semiring_Ol__distr,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( add_nat_Product_unit @ R3 @ X2 @ Y ) @ Z )
= ( add_nat_Product_unit @ R3 @ ( mult_n6028127365542633569t_unit @ R3 @ X2 @ Z ) @ ( mult_n6028127365542633569t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_846_semiring_Osemiring__simprules_I9_J,axiom,
! [R3: partia4934656038542163276t_unit,X2: set_int] :
( ( semiri8708897239777792527t_unit @ R3 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( mult_s3864001451298473021t_unit @ R3 @ ( one_se8065767436706823081t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_847_semiring_Osemiring__simprules_I9_J,axiom,
! [R3: partia4692342223508353374t_unit,X2: nat] :
( ( semiri3921172975686117281t_unit @ R3 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( mult_n6028127365542633569t_unit @ R3 @ ( one_na902338870878123981t_unit @ R3 ) @ X2 )
= X2 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_848_ring_Oup__smult__closed,axiom,
! [R3: partia4934656038542163276t_unit,A: set_int,P2: nat > set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_nat_set_int @ P2 @ ( up_set1168727741560211120t_unit @ R3 ) )
=> ( member_nat_set_int
@ ^ [I: nat] : ( mult_s3864001451298473021t_unit @ R3 @ A @ ( P2 @ I ) )
@ ( up_set1168727741560211120t_unit @ R3 ) ) ) ) ) ).
% ring.up_smult_closed
thf(fact_849_ring_Oup__smult__closed,axiom,
! [R3: partia4692342223508353374t_unit,A: nat,P2: nat > nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat_nat @ P2 @ ( up_nat_Product_unit @ R3 ) )
=> ( member_nat_nat
@ ^ [I: nat] : ( mult_n6028127365542633569t_unit @ R3 @ A @ ( P2 @ I ) )
@ ( up_nat_Product_unit @ R3 ) ) ) ) ) ).
% ring.up_smult_closed
thf(fact_850_abelian__group_Oa__coset__add__inv2,axiom,
! [G: partia4934656038542163276t_unit,M2: set_set_int,X2: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( ( a_r_co692709266861932262t_unit @ G @ M2 @ X2 )
= ( a_r_co692709266861932262t_unit @ G @ M2 @ Y ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( a_r_co692709266861932262t_unit @ G @ M2 @ ( add_se5859248395121729892t_unit @ G @ X2 @ ( a_inv_5951419416477254493t_unit @ G @ Y ) ) )
= M2 ) ) ) ) ) ) ).
% abelian_group.a_coset_add_inv2
thf(fact_851_abelian__group_Oa__coset__add__inv2,axiom,
! [G: partia4692342223508353374t_unit,M2: set_nat,X2: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( ( a_r_co9089071853028257912t_unit @ G @ M2 @ X2 )
= ( a_r_co9089071853028257912t_unit @ G @ M2 @ Y ) )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( a_r_co9089071853028257912t_unit @ G @ M2 @ ( add_nat_Product_unit @ G @ X2 @ ( a_inv_2472168910397739247t_unit @ G @ Y ) ) )
= M2 ) ) ) ) ) ) ).
% abelian_group.a_coset_add_inv2
thf(fact_852_abelian__group_Oa__coset__add__inv1,axiom,
! [G: partia4934656038542163276t_unit,M2: set_set_int,X2: set_int,Y: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( ( a_r_co692709266861932262t_unit @ G @ M2 @ ( add_se5859248395121729892t_unit @ G @ X2 @ ( a_inv_5951419416477254493t_unit @ G @ Y ) ) )
= M2 )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( ord_le4403425263959731960et_int @ M2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( a_r_co692709266861932262t_unit @ G @ M2 @ X2 )
= ( a_r_co692709266861932262t_unit @ G @ M2 @ Y ) ) ) ) ) ) ) ).
% abelian_group.a_coset_add_inv1
thf(fact_853_abelian__group_Oa__coset__add__inv1,axiom,
! [G: partia4692342223508353374t_unit,M2: set_nat,X2: nat,Y: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( ( a_r_co9089071853028257912t_unit @ G @ M2 @ ( add_nat_Product_unit @ G @ X2 @ ( a_inv_2472168910397739247t_unit @ G @ Y ) ) )
= M2 )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( ord_less_eq_set_nat @ M2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( a_r_co9089071853028257912t_unit @ G @ M2 @ X2 )
= ( a_r_co9089071853028257912t_unit @ G @ M2 @ Y ) ) ) ) ) ) ) ).
% abelian_group.a_coset_add_inv1
thf(fact_854_abelian__group_Oa__rcos__assoc__lcos,axiom,
! [G: partia4934656038542163276t_unit,H3: set_set_int,K: set_set_int,X2: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( set_ad273131178244904872t_unit @ G @ ( a_r_co692709266861932262t_unit @ G @ H3 @ X2 ) @ K )
= ( set_ad273131178244904872t_unit @ G @ H3 @ ( a_l_co3504123944629134560t_unit @ G @ X2 @ K ) ) ) ) ) ) ) ).
% abelian_group.a_rcos_assoc_lcos
thf(fact_855_abelian__group_Oa__rcos__assoc__lcos,axiom,
! [G: partia4692342223508353374t_unit,H3: set_nat,K: set_nat,X2: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( ord_less_eq_set_nat @ H3 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( ord_less_eq_set_nat @ K @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( set_ad320919470248169786t_unit @ G @ ( a_r_co9089071853028257912t_unit @ G @ H3 @ X2 ) @ K )
= ( set_ad320919470248169786t_unit @ G @ H3 @ ( a_l_co3340896127261556338t_unit @ G @ X2 @ K ) ) ) ) ) ) ) ).
% abelian_group.a_rcos_assoc_lcos
thf(fact_856_additive__subgroup_Ois__additive__subgroup,axiom,
! [H3: set_set_int,G: partia4934656038542163276t_unit] :
( ( additi7073586575563672860t_unit @ H3 @ G )
=> ( additi7073586575563672860t_unit @ H3 @ G ) ) ).
% additive_subgroup.is_additive_subgroup
thf(fact_857_ring_Oup__one__closed,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( member_nat_set_int
@ ^ [N4: nat] : ( if_set_int @ ( N4 = zero_zero_nat ) @ ( one_se8065767436706823081t_unit @ R3 ) @ ( zero_s6269048424454532197t_unit @ R3 ) )
@ ( up_set1168727741560211120t_unit @ R3 ) ) ) ).
% ring.up_one_closed
thf(fact_858_ring_Oline__extension__smult__closed,axiom,
! [R3: partia4692342223508353374t_unit,K: set_nat,E: set_nat,A: nat,K2: nat,U: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( subfie4892355163478727762t_unit @ K @ R3 )
=> ( ! [K3: nat,V2: nat] :
( ( member_nat @ K3 @ K )
=> ( ( member_nat @ V2 @ E )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R3 @ K3 @ V2 ) @ E ) ) )
=> ( ( ord_less_eq_set_nat @ E @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ K2 @ K )
=> ( ( member_nat @ U @ ( embedd838748496991043025t_unit @ R3 @ K @ A @ E ) )
=> ( member_nat @ ( mult_n6028127365542633569t_unit @ R3 @ K2 @ U ) @ ( embedd838748496991043025t_unit @ R3 @ K @ A @ E ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_859_ring_Oline__extension__smult__closed,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int,E: set_set_int,A: set_int,K2: set_int,U: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( subfie3888952257595785920t_unit @ K @ R3 )
=> ( ! [K3: set_int,V2: set_int] :
( ( member_set_int @ K3 @ K )
=> ( ( member_set_int @ V2 @ E )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R3 @ K3 @ V2 ) @ E ) ) )
=> ( ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ K2 @ K )
=> ( ( member_set_int @ U @ ( embedd4283282269743769663t_unit @ R3 @ K @ A @ E ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ R3 @ K2 @ U ) @ ( embedd4283282269743769663t_unit @ R3 @ K @ A @ E ) ) ) ) ) ) ) ) ) ).
% ring.line_extension_smult_closed
thf(fact_860_zfact__iso__inv__0,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( zfact_iso_inv @ N @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N ) ) ) )
= zero_zero_nat ) ) ).
% zfact_iso_inv_0
thf(fact_861_additive__subgroup_Oa__Hcarr,axiom,
! [H3: set_set_int,G: partia4934656038542163276t_unit,H: set_int] :
( ( additi7073586575563672860t_unit @ H3 @ G )
=> ( ( member_set_int @ H @ H3 )
=> ( member_set_int @ H @ ( partia966996272515721803t_unit @ G ) ) ) ) ).
% additive_subgroup.a_Hcarr
thf(fact_862_additive__subgroup_Oa__Hcarr,axiom,
! [H3: set_nat,G: partia4692342223508353374t_unit,H: nat] :
( ( additi4297497278381856430t_unit @ H3 @ G )
=> ( ( member_nat @ H @ H3 )
=> ( member_nat @ H @ ( partia3499330772048238685t_unit @ G ) ) ) ) ).
% additive_subgroup.a_Hcarr
thf(fact_863_additive__subgroup_Ozero__closed,axiom,
! [H3: set_set_int,G: partia4934656038542163276t_unit] :
( ( additi7073586575563672860t_unit @ H3 @ G )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ G ) @ H3 ) ) ).
% additive_subgroup.zero_closed
thf(fact_864_additive__subgroup_Oa__closed,axiom,
! [H3: set_set_int,G: partia4934656038542163276t_unit,X2: set_int,Y: set_int] :
( ( additi7073586575563672860t_unit @ H3 @ G )
=> ( ( member_set_int @ X2 @ H3 )
=> ( ( member_set_int @ Y @ H3 )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ G @ X2 @ Y ) @ H3 ) ) ) ) ).
% additive_subgroup.a_closed
thf(fact_865_additive__subgroup_Oa__inv__closed,axiom,
! [H3: set_set_int,G: partia4934656038542163276t_unit,X2: set_int] :
( ( additi7073586575563672860t_unit @ H3 @ G )
=> ( ( member_set_int @ X2 @ H3 )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ G @ X2 ) @ H3 ) ) ) ).
% additive_subgroup.a_inv_closed
thf(fact_866_additive__subgroup_Oa__subset,axiom,
! [H3: set_set_int,G: partia4934656038542163276t_unit] :
( ( additi7073586575563672860t_unit @ H3 @ G )
=> ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ G ) ) ) ).
% additive_subgroup.a_subset
thf(fact_867_additive__subgroup_Oa__subset,axiom,
! [H3: set_nat,G: partia4692342223508353374t_unit] :
( ( additi4297497278381856430t_unit @ H3 @ G )
=> ( ord_less_eq_set_nat @ H3 @ ( partia3499330772048238685t_unit @ G ) ) ) ).
% additive_subgroup.a_subset
thf(fact_868_abelian__group_Oadd__additive__subgroups,axiom,
! [G: partia4934656038542163276t_unit,H3: set_set_int,K: set_set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( additi7073586575563672860t_unit @ H3 @ G )
=> ( ( additi7073586575563672860t_unit @ K @ G )
=> ( additi7073586575563672860t_unit @ ( set_ad273131178244904872t_unit @ G @ H3 @ K ) @ G ) ) ) ) ).
% abelian_group.add_additive_subgroups
thf(fact_869_abelian__group_Oa__transpose__inv,axiom,
! [G: partia4934656038542163276t_unit,X2: set_int,Y: set_int,Z: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( ( add_se5859248395121729892t_unit @ G @ X2 @ Y )
= Z )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ G ) )
=> ( ( add_se5859248395121729892t_unit @ G @ ( a_inv_5951419416477254493t_unit @ G @ X2 ) @ Z )
= Y ) ) ) ) ) ) ).
% abelian_group.a_transpose_inv
thf(fact_870_abelian__group_Oa__transpose__inv,axiom,
! [G: partia4692342223508353374t_unit,X2: nat,Y: nat,Z: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( ( add_nat_Product_unit @ G @ X2 @ Y )
= Z )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ Z @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( add_nat_Product_unit @ G @ ( a_inv_2472168910397739247t_unit @ G @ X2 ) @ Z )
= Y ) ) ) ) ) ) ).
% abelian_group.a_transpose_inv
thf(fact_871_abelian__group_Osetadd__subset__G,axiom,
! [G: partia4934656038542163276t_unit,H3: set_set_int,K: set_set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ G ) )
=> ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ G @ H3 @ K ) @ ( partia966996272515721803t_unit @ G ) ) ) ) ) ).
% abelian_group.setadd_subset_G
thf(fact_872_abelian__group_Osetadd__subset__G,axiom,
! [G: partia4692342223508353374t_unit,H3: set_nat,K: set_nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( ord_less_eq_set_nat @ H3 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( ord_less_eq_set_nat @ K @ ( partia3499330772048238685t_unit @ G ) )
=> ( ord_less_eq_set_nat @ ( set_ad320919470248169786t_unit @ G @ H3 @ K ) @ ( partia3499330772048238685t_unit @ G ) ) ) ) ) ).
% abelian_group.setadd_subset_G
thf(fact_873_abelian__group_Oa__l__coset__subset__G,axiom,
! [G: partia4934656038542163276t_unit,H3: set_set_int,X2: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ord_le4403425263959731960et_int @ ( a_l_co3504123944629134560t_unit @ G @ X2 @ H3 ) @ ( partia966996272515721803t_unit @ G ) ) ) ) ) ).
% abelian_group.a_l_coset_subset_G
thf(fact_874_abelian__group_Oa__l__coset__subset__G,axiom,
! [G: partia4692342223508353374t_unit,H3: set_nat,X2: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( ord_less_eq_set_nat @ H3 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ord_less_eq_set_nat @ ( a_l_co3340896127261556338t_unit @ G @ X2 @ H3 ) @ ( partia3499330772048238685t_unit @ G ) ) ) ) ) ).
% abelian_group.a_l_coset_subset_G
thf(fact_875_abelian__monoid_Oset__add__closed,axiom,
! [G: partia4934656038542163276t_unit,A2: set_set_int,B3: set_set_int] :
( ( abelia3815030880812984441t_unit @ G )
=> ( ( ord_le4403425263959731960et_int @ A2 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( ord_le4403425263959731960et_int @ B3 @ ( partia966996272515721803t_unit @ G ) )
=> ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ G @ A2 @ B3 ) @ ( partia966996272515721803t_unit @ G ) ) ) ) ) ).
% abelian_monoid.set_add_closed
thf(fact_876_abelian__monoid_Oset__add__closed,axiom,
! [G: partia4692342223508353374t_unit,A2: set_nat,B3: set_nat] :
( ( abelia362511065248671243t_unit @ G )
=> ( ( ord_less_eq_set_nat @ A2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( ord_less_eq_set_nat @ B3 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ord_less_eq_set_nat @ ( set_ad320919470248169786t_unit @ G @ A2 @ B3 ) @ ( partia3499330772048238685t_unit @ G ) ) ) ) ) ).
% abelian_monoid.set_add_closed
thf(fact_877_subdomainI,axiom,
! [H3: set_set_int] :
( ( subcri1024317279029940167t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ! [H12: set_int,H22: set_int] :
( ( member_set_int @ H12 @ H3 )
=> ( ( member_set_int @ H22 @ H3 )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H12 @ H22 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( H12
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
| ( H22
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) )
=> ( subdom1520866149873910708t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% subdomainI
thf(fact_878_a__rcosetsI,axiom,
! [H3: set_set_int,X2: set_int] :
( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_set_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H3 @ X2 ) @ ( a_RCOS5559887075240879033t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H3 ) ) ) ) ).
% a_rcosetsI
thf(fact_879_monoid__cancelI,axiom,
( ! [A3: set_int,B2: set_int,C4: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C4 @ A3 )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C4 @ B2 ) )
=> ( ( member_set_int @ A3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: set_int,B2: set_int,C4: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A3 @ C4 )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B2 @ C4 ) )
=> ( ( member_set_int @ A3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid497721730651901107t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% monoid_cancelI
thf(fact_880_dividesI_H,axiom,
! [B: set_int,G: partia4934656038542163276t_unit,A: set_int,C: set_int] :
( ( B
= ( mult_s3864001451298473021t_unit @ G @ A @ C ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G ) )
=> ( factor5186451337065598620t_unit @ G @ A @ B ) ) ) ).
% dividesI'
thf(fact_881_dividesI_H,axiom,
! [B: nat,G: partia4692342223508353374t_unit,A: nat,C: nat] :
( ( B
= ( mult_n6028127365542633569t_unit @ G @ A @ C ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G ) )
=> ( factor7017787129603596992t_unit @ G @ A @ B ) ) ) ).
% dividesI'
thf(fact_882_subfield__m__inv__simprule,axiom,
! [K: set_set_int,K2: set_int,A: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ K2 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ A ) @ K )
=> ( member_set_int @ A @ K ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_883_DiffI,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ C @ A2 )
=> ( ~ ( member_nat_set_int @ C @ B3 )
=> ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_884_DiffI,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_885_DiffI,axiom,
! [C: set_int > nat,A2: set_set_int_nat,B3: set_set_int_nat] :
( ( member_set_int_nat @ C @ A2 )
=> ( ~ ( member_set_int_nat @ C @ B3 )
=> ( member_set_int_nat @ C @ ( minus_5256904771846099296nt_nat @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_886_DiffI,axiom,
! [C: set_set_int,A2: set_set_set_int,B3: set_set_set_int] :
( ( member_set_set_int @ C @ A2 )
=> ( ~ ( member_set_set_int @ C @ B3 )
=> ( member_set_set_int @ C @ ( minus_6857623457997529383et_int @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_887_DiffI,axiom,
! [C: set_int,A2: set_set_int,B3: set_set_int] :
( ( member_set_int @ C @ A2 )
=> ( ~ ( member_set_int @ C @ B3 )
=> ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_888_Diff__iff,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B3 ) )
= ( ( member_nat_set_int @ C @ A2 )
& ~ ( member_nat_set_int @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_889_Diff__iff,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_890_Diff__iff,axiom,
! [C: set_int > nat,A2: set_set_int_nat,B3: set_set_int_nat] :
( ( member_set_int_nat @ C @ ( minus_5256904771846099296nt_nat @ A2 @ B3 ) )
= ( ( member_set_int_nat @ C @ A2 )
& ~ ( member_set_int_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_891_Diff__iff,axiom,
! [C: set_set_int,A2: set_set_set_int,B3: set_set_set_int] :
( ( member_set_set_int @ C @ ( minus_6857623457997529383et_int @ A2 @ B3 ) )
= ( ( member_set_set_int @ C @ A2 )
& ~ ( member_set_set_int @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_892_Diff__iff,axiom,
! [C: set_int,A2: set_set_int,B3: set_set_int] :
( ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A2 @ B3 ) )
= ( ( member_set_int @ C @ A2 )
& ~ ( member_set_int @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_893_Diff__idemp,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( minus_8897228262479074673et_int @ ( minus_8897228262479074673et_int @ A2 @ B3 ) @ B3 )
= ( minus_8897228262479074673et_int @ A2 @ B3 ) ) ).
% Diff_idemp
thf(fact_894_Diff__cancel,axiom,
! [A2: set_set_set_int] :
( ( minus_6857623457997529383et_int @ A2 @ A2 )
= bot_bo2384636101374064866et_int ) ).
% Diff_cancel
thf(fact_895_Diff__cancel,axiom,
! [A2: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ A2 )
= bot_bot_set_set_int ) ).
% Diff_cancel
thf(fact_896_empty__Diff,axiom,
! [A2: set_set_set_int] :
( ( minus_6857623457997529383et_int @ bot_bo2384636101374064866et_int @ A2 )
= bot_bo2384636101374064866et_int ) ).
% empty_Diff
thf(fact_897_empty__Diff,axiom,
! [A2: set_set_int] :
( ( minus_8897228262479074673et_int @ bot_bot_set_set_int @ A2 )
= bot_bot_set_set_int ) ).
% empty_Diff
thf(fact_898_Diff__empty,axiom,
! [A2: set_set_set_int] :
( ( minus_6857623457997529383et_int @ A2 @ bot_bo2384636101374064866et_int )
= A2 ) ).
% Diff_empty
thf(fact_899_Diff__empty,axiom,
! [A2: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ bot_bot_set_set_int )
= A2 ) ).
% Diff_empty
thf(fact_900_insert__Diff1,axiom,
! [X2: nat > set_int,B3: set_nat_set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ X2 @ B3 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X2 @ A2 ) @ B3 )
= ( minus_3247115583872269408et_int @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_901_insert__Diff1,axiom,
! [X2: nat,B3: set_nat,A2: set_nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B3 )
= ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_902_insert__Diff1,axiom,
! [X2: set_int > nat,B3: set_set_int_nat,A2: set_set_int_nat] :
( ( member_set_int_nat @ X2 @ B3 )
=> ( ( minus_5256904771846099296nt_nat @ ( insert_set_int_nat @ X2 @ A2 ) @ B3 )
= ( minus_5256904771846099296nt_nat @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_903_insert__Diff1,axiom,
! [X2: set_set_int,B3: set_set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ X2 @ B3 )
=> ( ( minus_6857623457997529383et_int @ ( insert_set_set_int @ X2 @ A2 ) @ B3 )
= ( minus_6857623457997529383et_int @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_904_insert__Diff1,axiom,
! [X2: set_int,B3: set_set_int,A2: set_set_int] :
( ( member_set_int @ X2 @ B3 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X2 @ A2 ) @ B3 )
= ( minus_8897228262479074673et_int @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_905_Diff__insert0,axiom,
! [X2: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ~ ( member_nat_set_int @ X2 @ A2 )
=> ( ( minus_3247115583872269408et_int @ A2 @ ( insert_nat_set_int @ X2 @ B3 ) )
= ( minus_3247115583872269408et_int @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_906_Diff__insert0,axiom,
! [X2: nat,A2: set_nat,B3: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ B3 ) )
= ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_907_Diff__insert0,axiom,
! [X2: set_int > nat,A2: set_set_int_nat,B3: set_set_int_nat] :
( ~ ( member_set_int_nat @ X2 @ A2 )
=> ( ( minus_5256904771846099296nt_nat @ A2 @ ( insert_set_int_nat @ X2 @ B3 ) )
= ( minus_5256904771846099296nt_nat @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_908_Diff__insert0,axiom,
! [X2: set_set_int,A2: set_set_set_int,B3: set_set_set_int] :
( ~ ( member_set_set_int @ X2 @ A2 )
=> ( ( minus_6857623457997529383et_int @ A2 @ ( insert_set_set_int @ X2 @ B3 ) )
= ( minus_6857623457997529383et_int @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_909_Diff__insert0,axiom,
! [X2: set_int,A2: set_set_int,B3: set_set_int] :
( ~ ( member_set_int @ X2 @ A2 )
=> ( ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ X2 @ B3 ) )
= ( minus_8897228262479074673et_int @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_910_Diff__eq__empty__iff,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( ( minus_8897228262479074673et_int @ A2 @ B3 )
= bot_bot_set_set_int )
= ( ord_le4403425263959731960et_int @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_911_Diff__eq__empty__iff,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ( ( minus_6857623457997529383et_int @ A2 @ B3 )
= bot_bo2384636101374064866et_int )
= ( ord_le4317611570275147438et_int @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_912_insert__Diff__single,axiom,
! [A: set_set_int,A2: set_set_set_int] :
( ( insert_set_set_int @ A @ ( minus_6857623457997529383et_int @ A2 @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) )
= ( insert_set_set_int @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_913_insert__Diff__single,axiom,
! [A: set_int,A2: set_set_int] :
( ( insert_set_int @ A @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) )
= ( insert_set_int @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_914_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_915_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_916_DiffE,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B3 ) )
=> ~ ( ( member_nat_set_int @ C @ A2 )
=> ( member_nat_set_int @ C @ B3 ) ) ) ).
% DiffE
thf(fact_917_DiffE,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_918_DiffE,axiom,
! [C: set_int > nat,A2: set_set_int_nat,B3: set_set_int_nat] :
( ( member_set_int_nat @ C @ ( minus_5256904771846099296nt_nat @ A2 @ B3 ) )
=> ~ ( ( member_set_int_nat @ C @ A2 )
=> ( member_set_int_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_919_DiffE,axiom,
! [C: set_set_int,A2: set_set_set_int,B3: set_set_set_int] :
( ( member_set_set_int @ C @ ( minus_6857623457997529383et_int @ A2 @ B3 ) )
=> ~ ( ( member_set_set_int @ C @ A2 )
=> ( member_set_set_int @ C @ B3 ) ) ) ).
% DiffE
thf(fact_920_DiffE,axiom,
! [C: set_int,A2: set_set_int,B3: set_set_int] :
( ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A2 @ B3 ) )
=> ~ ( ( member_set_int @ C @ A2 )
=> ( member_set_int @ C @ B3 ) ) ) ).
% DiffE
thf(fact_921_DiffD1,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B3 ) )
=> ( member_nat_set_int @ C @ A2 ) ) ).
% DiffD1
thf(fact_922_DiffD1,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_923_DiffD1,axiom,
! [C: set_int > nat,A2: set_set_int_nat,B3: set_set_int_nat] :
( ( member_set_int_nat @ C @ ( minus_5256904771846099296nt_nat @ A2 @ B3 ) )
=> ( member_set_int_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_924_DiffD1,axiom,
! [C: set_set_int,A2: set_set_set_int,B3: set_set_set_int] :
( ( member_set_set_int @ C @ ( minus_6857623457997529383et_int @ A2 @ B3 ) )
=> ( member_set_set_int @ C @ A2 ) ) ).
% DiffD1
thf(fact_925_DiffD1,axiom,
! [C: set_int,A2: set_set_int,B3: set_set_int] :
( ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A2 @ B3 ) )
=> ( member_set_int @ C @ A2 ) ) ).
% DiffD1
thf(fact_926_DiffD2,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B3: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B3 ) )
=> ~ ( member_nat_set_int @ C @ B3 ) ) ).
% DiffD2
thf(fact_927_DiffD2,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
=> ~ ( member_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_928_DiffD2,axiom,
! [C: set_int > nat,A2: set_set_int_nat,B3: set_set_int_nat] :
( ( member_set_int_nat @ C @ ( minus_5256904771846099296nt_nat @ A2 @ B3 ) )
=> ~ ( member_set_int_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_929_DiffD2,axiom,
! [C: set_set_int,A2: set_set_set_int,B3: set_set_set_int] :
( ( member_set_set_int @ C @ ( minus_6857623457997529383et_int @ A2 @ B3 ) )
=> ~ ( member_set_set_int @ C @ B3 ) ) ).
% DiffD2
thf(fact_930_DiffD2,axiom,
! [C: set_int,A2: set_set_int,B3: set_set_int] :
( ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A2 @ B3 ) )
=> ~ ( member_set_int @ C @ B3 ) ) ).
% DiffD2
thf(fact_931_set__diff__eq,axiom,
( minus_3247115583872269408et_int
= ( ^ [A4: set_nat_set_int,B4: set_nat_set_int] :
( collect_nat_set_int
@ ^ [X4: nat > set_int] :
( ( member_nat_set_int @ X4 @ A4 )
& ~ ( member_nat_set_int @ X4 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_932_set__diff__eq,axiom,
( minus_5256904771846099296nt_nat
= ( ^ [A4: set_set_int_nat,B4: set_set_int_nat] :
( collect_set_int_nat
@ ^ [X4: set_int > nat] :
( ( member_set_int_nat @ X4 @ A4 )
& ~ ( member_set_int_nat @ X4 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_933_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ~ ( member_nat @ X4 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_934_set__diff__eq,axiom,
( minus_6857623457997529383et_int
= ( ^ [A4: set_set_set_int,B4: set_set_set_int] :
( collect_set_set_int
@ ^ [X4: set_set_int] :
( ( member_set_set_int @ X4 @ A4 )
& ~ ( member_set_set_int @ X4 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_935_set__diff__eq,axiom,
( minus_8897228262479074673et_int
= ( ^ [A4: set_set_int,B4: set_set_int] :
( collect_set_int
@ ^ [X4: set_int] :
( ( member_set_int @ X4 @ A4 )
& ~ ( member_set_int @ X4 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_936_double__diff,axiom,
! [A2: set_set_int,B3: set_set_int,C2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ( ( ord_le4403425263959731960et_int @ B3 @ C2 )
=> ( ( minus_8897228262479074673et_int @ B3 @ ( minus_8897228262479074673et_int @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_937_double__diff,axiom,
! [A2: set_set_set_int,B3: set_set_set_int,C2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ( ord_le4317611570275147438et_int @ B3 @ C2 )
=> ( ( minus_6857623457997529383et_int @ B3 @ ( minus_6857623457997529383et_int @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_938_Diff__subset,axiom,
! [A2: set_set_int,B3: set_set_int] : ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_939_Diff__subset,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] : ( ord_le4317611570275147438et_int @ ( minus_6857623457997529383et_int @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_940_Diff__mono,axiom,
! [A2: set_set_int,C2: set_set_int,D: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ C2 )
=> ( ( ord_le4403425263959731960et_int @ D @ B3 )
=> ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A2 @ B3 ) @ ( minus_8897228262479074673et_int @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_941_Diff__mono,axiom,
! [A2: set_set_set_int,C2: set_set_set_int,D: set_set_set_int,B3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ C2 )
=> ( ( ord_le4317611570275147438et_int @ D @ B3 )
=> ( ord_le4317611570275147438et_int @ ( minus_6857623457997529383et_int @ A2 @ B3 ) @ ( minus_6857623457997529383et_int @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_942_insert__Diff__if,axiom,
! [X2: nat > set_int,B3: set_nat_set_int,A2: set_nat_set_int] :
( ( ( member_nat_set_int @ X2 @ B3 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X2 @ A2 ) @ B3 )
= ( minus_3247115583872269408et_int @ A2 @ B3 ) ) )
& ( ~ ( member_nat_set_int @ X2 @ B3 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X2 @ A2 ) @ B3 )
= ( insert_nat_set_int @ X2 @ ( minus_3247115583872269408et_int @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_943_insert__Diff__if,axiom,
! [X2: nat,B3: set_nat,A2: set_nat] :
( ( ( member_nat @ X2 @ B3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B3 )
= ( minus_minus_set_nat @ A2 @ B3 ) ) )
& ( ~ ( member_nat @ X2 @ B3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B3 )
= ( insert_nat @ X2 @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_944_insert__Diff__if,axiom,
! [X2: set_int > nat,B3: set_set_int_nat,A2: set_set_int_nat] :
( ( ( member_set_int_nat @ X2 @ B3 )
=> ( ( minus_5256904771846099296nt_nat @ ( insert_set_int_nat @ X2 @ A2 ) @ B3 )
= ( minus_5256904771846099296nt_nat @ A2 @ B3 ) ) )
& ( ~ ( member_set_int_nat @ X2 @ B3 )
=> ( ( minus_5256904771846099296nt_nat @ ( insert_set_int_nat @ X2 @ A2 ) @ B3 )
= ( insert_set_int_nat @ X2 @ ( minus_5256904771846099296nt_nat @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_945_insert__Diff__if,axiom,
! [X2: set_set_int,B3: set_set_set_int,A2: set_set_set_int] :
( ( ( member_set_set_int @ X2 @ B3 )
=> ( ( minus_6857623457997529383et_int @ ( insert_set_set_int @ X2 @ A2 ) @ B3 )
= ( minus_6857623457997529383et_int @ A2 @ B3 ) ) )
& ( ~ ( member_set_set_int @ X2 @ B3 )
=> ( ( minus_6857623457997529383et_int @ ( insert_set_set_int @ X2 @ A2 ) @ B3 )
= ( insert_set_set_int @ X2 @ ( minus_6857623457997529383et_int @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_946_insert__Diff__if,axiom,
! [X2: set_int,B3: set_set_int,A2: set_set_int] :
( ( ( member_set_int @ X2 @ B3 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X2 @ A2 ) @ B3 )
= ( minus_8897228262479074673et_int @ A2 @ B3 ) ) )
& ( ~ ( member_set_int @ X2 @ B3 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X2 @ A2 ) @ B3 )
= ( insert_set_int @ X2 @ ( minus_8897228262479074673et_int @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_947_psubset__imp__ex__mem,axiom,
! [A2: set_nat_set_int,B3: set_nat_set_int] :
( ( ord_le2931775347370382171et_int @ A2 @ B3 )
=> ? [B2: nat > set_int] : ( member_nat_set_int @ B2 @ ( minus_3247115583872269408et_int @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_948_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ? [B2: nat] : ( member_nat @ B2 @ ( minus_minus_set_nat @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_949_psubset__imp__ex__mem,axiom,
! [A2: set_set_int_nat,B3: set_set_int_nat] :
( ( ord_le4941564535344212059nt_nat @ A2 @ B3 )
=> ? [B2: set_int > nat] : ( member_set_int_nat @ B2 @ ( minus_5256904771846099296nt_nat @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_950_psubset__imp__ex__mem,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ( ord_le4562804192517611682et_int @ A2 @ B3 )
=> ? [B2: set_set_int] : ( member_set_set_int @ B2 @ ( minus_6857623457997529383et_int @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_951_psubset__imp__ex__mem,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B3 )
=> ? [B2: set_int] : ( member_set_int @ B2 @ ( minus_8897228262479074673et_int @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_952_monoid__cancel_Ol__cancel,axiom,
! [G: partia4934656038542163276t_unit,C: set_int,A: set_int,B: set_int] :
( ( monoid497721730651901107t_unit @ G )
=> ( ( ( mult_s3864001451298473021t_unit @ G @ C @ A )
= ( mult_s3864001451298473021t_unit @ G @ C @ B ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_953_monoid__cancel_Ol__cancel,axiom,
! [G: partia4692342223508353374t_unit,C: nat,A: nat,B: nat] :
( ( monoid2713784563552164055t_unit @ G )
=> ( ( ( mult_n6028127365542633569t_unit @ G @ C @ A )
= ( mult_n6028127365542633569t_unit @ G @ C @ B ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_954_monoid__cancel_Or__cancel,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,C: set_int,B: set_int] :
( ( monoid497721730651901107t_unit @ G )
=> ( ( ( mult_s3864001451298473021t_unit @ G @ A @ C )
= ( mult_s3864001451298473021t_unit @ G @ B @ C ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_955_monoid__cancel_Or__cancel,axiom,
! [G: partia4692342223508353374t_unit,A: nat,C: nat,B: nat] :
( ( monoid2713784563552164055t_unit @ G )
=> ( ( ( mult_n6028127365542633569t_unit @ G @ A @ C )
= ( mult_n6028127365542633569t_unit @ G @ B @ C ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_956_subset__Diff__insert,axiom,
! [A2: set_nat_set_int,B3: set_nat_set_int,X2: nat > set_int,C2: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ ( minus_3247115583872269408et_int @ B3 @ ( insert_nat_set_int @ X2 @ C2 ) ) )
= ( ( ord_le5995675665013768039et_int @ A2 @ ( minus_3247115583872269408et_int @ B3 @ C2 ) )
& ~ ( member_nat_set_int @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_957_subset__Diff__insert,axiom,
! [A2: set_nat,B3: set_nat,X2: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B3 @ ( insert_nat @ X2 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B3 @ C2 ) )
& ~ ( member_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_958_subset__Diff__insert,axiom,
! [A2: set_set_int_nat,B3: set_set_int_nat,X2: set_int > nat,C2: set_set_int_nat] :
( ( ord_le8005464852987597927nt_nat @ A2 @ ( minus_5256904771846099296nt_nat @ B3 @ ( insert_set_int_nat @ X2 @ C2 ) ) )
= ( ( ord_le8005464852987597927nt_nat @ A2 @ ( minus_5256904771846099296nt_nat @ B3 @ C2 ) )
& ~ ( member_set_int_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_959_subset__Diff__insert,axiom,
! [A2: set_set_int,B3: set_set_int,X2: set_int,C2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( minus_8897228262479074673et_int @ B3 @ ( insert_set_int @ X2 @ C2 ) ) )
= ( ( ord_le4403425263959731960et_int @ A2 @ ( minus_8897228262479074673et_int @ B3 @ C2 ) )
& ~ ( member_set_int @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_960_subset__Diff__insert,axiom,
! [A2: set_set_set_int,B3: set_set_set_int,X2: set_set_int,C2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ ( minus_6857623457997529383et_int @ B3 @ ( insert_set_set_int @ X2 @ C2 ) ) )
= ( ( ord_le4317611570275147438et_int @ A2 @ ( minus_6857623457997529383et_int @ B3 @ C2 ) )
& ~ ( member_set_set_int @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_961_Diff__insert__absorb,axiom,
! [X2: nat > set_int,A2: set_nat_set_int] :
( ~ ( member_nat_set_int @ X2 @ A2 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X2 @ A2 ) @ ( insert_nat_set_int @ X2 @ bot_bo8417611410066262939et_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_962_Diff__insert__absorb,axiom,
! [X2: nat,A2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_963_Diff__insert__absorb,axiom,
! [X2: set_int > nat,A2: set_set_int_nat] :
( ~ ( member_set_int_nat @ X2 @ A2 )
=> ( ( minus_5256904771846099296nt_nat @ ( insert_set_int_nat @ X2 @ A2 ) @ ( insert_set_int_nat @ X2 @ bot_bo1204028561185317019nt_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_964_Diff__insert__absorb,axiom,
! [X2: set_set_int,A2: set_set_set_int] :
( ~ ( member_set_set_int @ X2 @ A2 )
=> ( ( minus_6857623457997529383et_int @ ( insert_set_set_int @ X2 @ A2 ) @ ( insert_set_set_int @ X2 @ bot_bo2384636101374064866et_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_965_Diff__insert__absorb,axiom,
! [X2: set_int,A2: set_set_int] :
( ~ ( member_set_int @ X2 @ A2 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X2 @ A2 ) @ ( insert_set_int @ X2 @ bot_bot_set_set_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_966_Diff__insert2,axiom,
! [A2: set_set_set_int,A: set_set_int,B3: set_set_set_int] :
( ( minus_6857623457997529383et_int @ A2 @ ( insert_set_set_int @ A @ B3 ) )
= ( minus_6857623457997529383et_int @ ( minus_6857623457997529383et_int @ A2 @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_967_Diff__insert2,axiom,
! [A2: set_set_int,A: set_int,B3: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ B3 ) )
= ( minus_8897228262479074673et_int @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_968_insert__Diff,axiom,
! [A: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ A2 )
=> ( ( insert_nat_set_int @ A @ ( minus_3247115583872269408et_int @ A2 @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_969_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_970_insert__Diff,axiom,
! [A: set_int > nat,A2: set_set_int_nat] :
( ( member_set_int_nat @ A @ A2 )
=> ( ( insert_set_int_nat @ A @ ( minus_5256904771846099296nt_nat @ A2 @ ( insert_set_int_nat @ A @ bot_bo1204028561185317019nt_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_971_insert__Diff,axiom,
! [A: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ A @ A2 )
=> ( ( insert_set_set_int @ A @ ( minus_6857623457997529383et_int @ A2 @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_972_insert__Diff,axiom,
! [A: set_int,A2: set_set_int] :
( ( member_set_int @ A @ A2 )
=> ( ( insert_set_int @ A @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_973_Diff__insert,axiom,
! [A2: set_set_set_int,A: set_set_int,B3: set_set_set_int] :
( ( minus_6857623457997529383et_int @ A2 @ ( insert_set_set_int @ A @ B3 ) )
= ( minus_6857623457997529383et_int @ ( minus_6857623457997529383et_int @ A2 @ B3 ) @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) ) ).
% Diff_insert
thf(fact_974_Diff__insert,axiom,
! [A2: set_set_int,A: set_int,B3: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ B3 ) )
= ( minus_8897228262479074673et_int @ ( minus_8897228262479074673et_int @ A2 @ B3 ) @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ) ).
% Diff_insert
thf(fact_975_monoid__cancel_Odivides__mult__l,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int,C: set_int] :
( ( monoid497721730651901107t_unit @ G )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G ) )
=> ( ( factor5186451337065598620t_unit @ G @ ( mult_s3864001451298473021t_unit @ G @ C @ A ) @ ( mult_s3864001451298473021t_unit @ G @ C @ B ) )
= ( factor5186451337065598620t_unit @ G @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_976_monoid__cancel_Odivides__mult__l,axiom,
! [G: partia4692342223508353374t_unit,A: nat,B: nat,C: nat] :
( ( monoid2713784563552164055t_unit @ G )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( factor7017787129603596992t_unit @ G @ ( mult_n6028127365542633569t_unit @ G @ C @ A ) @ ( mult_n6028127365542633569t_unit @ G @ C @ B ) )
= ( factor7017787129603596992t_unit @ G @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.divides_mult_l
thf(fact_977_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int,B6: set_int] :
( ( monoid497721730651901107t_unit @ G )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ B6 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( associ3816808824722140549t_unit @ G @ ( mult_s3864001451298473021t_unit @ G @ A @ B ) @ ( mult_s3864001451298473021t_unit @ G @ A @ B6 ) )
=> ( associ3816808824722140549t_unit @ G @ B @ B6 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_978_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia4692342223508353374t_unit,A: nat,B: nat,B6: nat] :
( ( monoid2713784563552164055t_unit @ G )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ B6 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( associ4357985397585971625t_unit @ G @ ( mult_n6028127365542633569t_unit @ G @ A @ B ) @ ( mult_n6028127365542633569t_unit @ G @ A @ B6 ) )
=> ( associ4357985397585971625t_unit @ G @ B @ B6 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_979_monoid__cancel_Oproperfactor__mult__l,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int,C: set_int] :
( ( monoid497721730651901107t_unit @ G )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G ) )
=> ( ( proper1002977052347345036t_unit @ G @ ( mult_s3864001451298473021t_unit @ G @ C @ A ) @ ( mult_s3864001451298473021t_unit @ G @ C @ B ) )
= ( proper1002977052347345036t_unit @ G @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_l
thf(fact_980_monoid__cancel_Oproperfactor__mult__l,axiom,
! [G: partia4692342223508353374t_unit,A: nat,B: nat,C: nat] :
( ( monoid2713784563552164055t_unit @ G )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( proper2699154196232879280t_unit @ G @ ( mult_n6028127365542633569t_unit @ G @ C @ A ) @ ( mult_n6028127365542633569t_unit @ G @ C @ B ) )
= ( proper2699154196232879280t_unit @ G @ A @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_l
thf(fact_981_monoid__cancel_Oproperfactor__mult__lI,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int,C: set_int] :
( ( monoid497721730651901107t_unit @ G )
=> ( ( proper1002977052347345036t_unit @ G @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G ) )
=> ( proper1002977052347345036t_unit @ G @ ( mult_s3864001451298473021t_unit @ G @ C @ A ) @ ( mult_s3864001451298473021t_unit @ G @ C @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_lI
thf(fact_982_monoid__cancel_Oproperfactor__mult__lI,axiom,
! [G: partia4692342223508353374t_unit,A: nat,B: nat,C: nat] :
( ( monoid2713784563552164055t_unit @ G )
=> ( ( proper2699154196232879280t_unit @ G @ A @ B )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G ) )
=> ( proper2699154196232879280t_unit @ G @ ( mult_n6028127365542633569t_unit @ G @ C @ A ) @ ( mult_n6028127365542633569t_unit @ G @ C @ B ) ) ) ) ) ) ).
% monoid_cancel.properfactor_mult_lI
thf(fact_983_subset__insert__iff,axiom,
! [A2: set_nat_set_int,X2: nat > set_int,B3: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ ( insert_nat_set_int @ X2 @ B3 ) )
= ( ( ( member_nat_set_int @ X2 @ A2 )
=> ( ord_le5995675665013768039et_int @ ( minus_3247115583872269408et_int @ A2 @ ( insert_nat_set_int @ X2 @ bot_bo8417611410066262939et_int ) ) @ B3 ) )
& ( ~ ( member_nat_set_int @ X2 @ A2 )
=> ( ord_le5995675665013768039et_int @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_984_subset__insert__iff,axiom,
! [A2: set_nat,X2: nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B3 ) )
= ( ( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B3 ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_985_subset__insert__iff,axiom,
! [A2: set_set_int_nat,X2: set_int > nat,B3: set_set_int_nat] :
( ( ord_le8005464852987597927nt_nat @ A2 @ ( insert_set_int_nat @ X2 @ B3 ) )
= ( ( ( member_set_int_nat @ X2 @ A2 )
=> ( ord_le8005464852987597927nt_nat @ ( minus_5256904771846099296nt_nat @ A2 @ ( insert_set_int_nat @ X2 @ bot_bo1204028561185317019nt_nat ) ) @ B3 ) )
& ( ~ ( member_set_int_nat @ X2 @ A2 )
=> ( ord_le8005464852987597927nt_nat @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_986_subset__insert__iff,axiom,
! [A2: set_set_int,X2: set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X2 @ B3 ) )
= ( ( ( member_set_int @ X2 @ A2 )
=> ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ X2 @ bot_bot_set_set_int ) ) @ B3 ) )
& ( ~ ( member_set_int @ X2 @ A2 )
=> ( ord_le4403425263959731960et_int @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_987_subset__insert__iff,axiom,
! [A2: set_set_set_int,X2: set_set_int,B3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ X2 @ B3 ) )
= ( ( ( member_set_set_int @ X2 @ A2 )
=> ( ord_le4317611570275147438et_int @ ( minus_6857623457997529383et_int @ A2 @ ( insert_set_set_int @ X2 @ bot_bo2384636101374064866et_int ) ) @ B3 ) )
& ( ~ ( member_set_set_int @ X2 @ A2 )
=> ( ord_le4317611570275147438et_int @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_988_Diff__single__insert,axiom,
! [A2: set_set_int,X2: set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ X2 @ bot_bot_set_set_int ) ) @ B3 )
=> ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X2 @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_989_Diff__single__insert,axiom,
! [A2: set_set_set_int,X2: set_set_int,B3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ ( minus_6857623457997529383et_int @ A2 @ ( insert_set_set_int @ X2 @ bot_bo2384636101374064866et_int ) ) @ B3 )
=> ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ X2 @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_990_psubset__insert__iff,axiom,
! [A2: set_nat_set_int,X2: nat > set_int,B3: set_nat_set_int] :
( ( ord_le2931775347370382171et_int @ A2 @ ( insert_nat_set_int @ X2 @ B3 ) )
= ( ( ( member_nat_set_int @ X2 @ B3 )
=> ( ord_le2931775347370382171et_int @ A2 @ B3 ) )
& ( ~ ( member_nat_set_int @ X2 @ B3 )
=> ( ( ( member_nat_set_int @ X2 @ A2 )
=> ( ord_le2931775347370382171et_int @ ( minus_3247115583872269408et_int @ A2 @ ( insert_nat_set_int @ X2 @ bot_bo8417611410066262939et_int ) ) @ B3 ) )
& ( ~ ( member_nat_set_int @ X2 @ A2 )
=> ( ord_le5995675665013768039et_int @ A2 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_991_psubset__insert__iff,axiom,
! [A2: set_nat,X2: nat,B3: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat @ X2 @ B3 ) )
= ( ( ( member_nat @ X2 @ B3 )
=> ( ord_less_set_nat @ A2 @ B3 ) )
& ( ~ ( member_nat @ X2 @ B3 )
=> ( ( ( member_nat @ X2 @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B3 ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_992_psubset__insert__iff,axiom,
! [A2: set_set_int_nat,X2: set_int > nat,B3: set_set_int_nat] :
( ( ord_le4941564535344212059nt_nat @ A2 @ ( insert_set_int_nat @ X2 @ B3 ) )
= ( ( ( member_set_int_nat @ X2 @ B3 )
=> ( ord_le4941564535344212059nt_nat @ A2 @ B3 ) )
& ( ~ ( member_set_int_nat @ X2 @ B3 )
=> ( ( ( member_set_int_nat @ X2 @ A2 )
=> ( ord_le4941564535344212059nt_nat @ ( minus_5256904771846099296nt_nat @ A2 @ ( insert_set_int_nat @ X2 @ bot_bo1204028561185317019nt_nat ) ) @ B3 ) )
& ( ~ ( member_set_int_nat @ X2 @ A2 )
=> ( ord_le8005464852987597927nt_nat @ A2 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_993_psubset__insert__iff,axiom,
! [A2: set_set_int,X2: set_int,B3: set_set_int] :
( ( ord_less_set_set_int @ A2 @ ( insert_set_int @ X2 @ B3 ) )
= ( ( ( member_set_int @ X2 @ B3 )
=> ( ord_less_set_set_int @ A2 @ B3 ) )
& ( ~ ( member_set_int @ X2 @ B3 )
=> ( ( ( member_set_int @ X2 @ A2 )
=> ( ord_less_set_set_int @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ X2 @ bot_bot_set_set_int ) ) @ B3 ) )
& ( ~ ( member_set_int @ X2 @ A2 )
=> ( ord_le4403425263959731960et_int @ A2 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_994_psubset__insert__iff,axiom,
! [A2: set_set_set_int,X2: set_set_int,B3: set_set_set_int] :
( ( ord_le4562804192517611682et_int @ A2 @ ( insert_set_set_int @ X2 @ B3 ) )
= ( ( ( member_set_set_int @ X2 @ B3 )
=> ( ord_le4562804192517611682et_int @ A2 @ B3 ) )
& ( ~ ( member_set_set_int @ X2 @ B3 )
=> ( ( ( member_set_set_int @ X2 @ A2 )
=> ( ord_le4562804192517611682et_int @ ( minus_6857623457997529383et_int @ A2 @ ( insert_set_set_int @ X2 @ bot_bo2384636101374064866et_int ) ) @ B3 ) )
& ( ~ ( member_set_set_int @ X2 @ A2 )
=> ( ord_le4317611570275147438et_int @ A2 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_995_abelian__group_Oa__rcosetsI,axiom,
! [G: partia4934656038542163276t_unit,H3: set_set_int,X2: set_int] :
( ( abelia23968383328945916t_unit @ G )
=> ( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ G ) )
=> ( member_set_set_int @ ( a_r_co692709266861932262t_unit @ G @ H3 @ X2 ) @ ( a_RCOS5559887075240879033t_unit @ G @ H3 ) ) ) ) ) ).
% abelian_group.a_rcosetsI
thf(fact_996_abelian__group_Oa__rcosetsI,axiom,
! [G: partia4692342223508353374t_unit,H3: set_nat,X2: nat] :
( ( abelia406319425121669262t_unit @ G )
=> ( ( ord_less_eq_set_nat @ H3 @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ G ) )
=> ( member_set_nat @ ( a_r_co9089071853028257912t_unit @ G @ H3 @ X2 ) @ ( a_RCOS6328597822339572043t_unit @ G @ H3 ) ) ) ) ) ).
% abelian_group.a_rcosetsI
thf(fact_997_divides__antisym,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( factor5186451337065598620t_unit @ G @ A @ B )
=> ( ( factor5186451337065598620t_unit @ G @ B @ A )
=> ( associ3816808824722140549t_unit @ G @ A @ B ) ) ) ).
% divides_antisym
thf(fact_998_associated__def,axiom,
( associ3816808824722140549t_unit
= ( ^ [G3: partia4934656038542163276t_unit,A5: set_int,B7: set_int] :
( ( factor5186451337065598620t_unit @ G3 @ A5 @ B7 )
& ( factor5186451337065598620t_unit @ G3 @ B7 @ A5 ) ) ) ) ).
% associated_def
thf(fact_999_associatedE,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( associ3816808824722140549t_unit @ G @ A @ B )
=> ~ ( ( factor5186451337065598620t_unit @ G @ A @ B )
=> ~ ( factor5186451337065598620t_unit @ G @ B @ A ) ) ) ).
% associatedE
thf(fact_1000_associatedD,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( associ3816808824722140549t_unit @ G @ A @ B )
=> ( factor5186451337065598620t_unit @ G @ A @ B ) ) ).
% associatedD
thf(fact_1001_properfactor__def,axiom,
( proper1002977052347345036t_unit
= ( ^ [G3: partia4934656038542163276t_unit,A5: set_int,B7: set_int] :
( ( factor5186451337065598620t_unit @ G3 @ A5 @ B7 )
& ~ ( factor5186451337065598620t_unit @ G3 @ B7 @ A5 ) ) ) ) ).
% properfactor_def
thf(fact_1002_properfactorI,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( factor5186451337065598620t_unit @ G @ A @ B )
=> ( ~ ( factor5186451337065598620t_unit @ G @ B @ A )
=> ( proper1002977052347345036t_unit @ G @ A @ B ) ) ) ).
% properfactorI
thf(fact_1003_properfactorE,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( proper1002977052347345036t_unit @ G @ A @ B )
=> ~ ( ( factor5186451337065598620t_unit @ G @ A @ B )
=> ( factor5186451337065598620t_unit @ G @ B @ A ) ) ) ).
% properfactorE
thf(fact_1004_dividesD,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( factor5186451337065598620t_unit @ G @ A @ B )
=> ? [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ G ) )
& ( B
= ( mult_s3864001451298473021t_unit @ G @ A @ X ) ) ) ) ).
% dividesD
thf(fact_1005_dividesD,axiom,
! [G: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( factor7017787129603596992t_unit @ G @ A @ B )
=> ? [X: nat] :
( ( member_nat @ X @ ( partia3499330772048238685t_unit @ G ) )
& ( B
= ( mult_n6028127365542633569t_unit @ G @ A @ X ) ) ) ) ).
% dividesD
thf(fact_1006_dividesE,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( factor5186451337065598620t_unit @ G @ A @ B )
=> ~ ! [C4: set_int] :
( ( B
= ( mult_s3864001451298473021t_unit @ G @ A @ C4 ) )
=> ~ ( member_set_int @ C4 @ ( partia966996272515721803t_unit @ G ) ) ) ) ).
% dividesE
thf(fact_1007_dividesE,axiom,
! [G: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( factor7017787129603596992t_unit @ G @ A @ B )
=> ~ ! [C4: nat] :
( ( B
= ( mult_n6028127365542633569t_unit @ G @ A @ C4 ) )
=> ~ ( member_nat @ C4 @ ( partia3499330772048238685t_unit @ G ) ) ) ) ).
% dividesE
thf(fact_1008_dividesI,axiom,
! [C: set_int,G: partia4934656038542163276t_unit,B: set_int,A: set_int] :
( ( member_set_int @ C @ ( partia966996272515721803t_unit @ G ) )
=> ( ( B
= ( mult_s3864001451298473021t_unit @ G @ A @ C ) )
=> ( factor5186451337065598620t_unit @ G @ A @ B ) ) ) ).
% dividesI
thf(fact_1009_dividesI,axiom,
! [C: nat,G: partia4692342223508353374t_unit,B: nat,A: nat] :
( ( member_nat @ C @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( B
= ( mult_n6028127365542633569t_unit @ G @ A @ C ) )
=> ( factor7017787129603596992t_unit @ G @ A @ B ) ) ) ).
% dividesI
thf(fact_1010_factor__def,axiom,
( factor5186451337065598620t_unit
= ( ^ [G3: partia4934656038542163276t_unit,A5: set_int,B7: set_int] :
? [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ G3 ) )
& ( B7
= ( mult_s3864001451298473021t_unit @ G3 @ A5 @ X4 ) ) ) ) ) ).
% factor_def
thf(fact_1011_factor__def,axiom,
( factor7017787129603596992t_unit
= ( ^ [G3: partia4692342223508353374t_unit,A5: nat,B7: nat] :
? [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ G3 ) )
& ( B7
= ( mult_n6028127365542633569t_unit @ G3 @ A5 @ X4 ) ) ) ) ) ).
% factor_def
thf(fact_1012_properfactorI2,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( factor5186451337065598620t_unit @ G @ A @ B )
=> ( ~ ( associ3816808824722140549t_unit @ G @ A @ B )
=> ( proper1002977052347345036t_unit @ G @ A @ B ) ) ) ).
% properfactorI2
thf(fact_1013_properfactorE2,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( proper1002977052347345036t_unit @ G @ A @ B )
=> ~ ( ( factor5186451337065598620t_unit @ G @ A @ B )
=> ( associ3816808824722140549t_unit @ G @ A @ B ) ) ) ).
% properfactorE2
thf(fact_1014_isgcd__def,axiom,
( isgcd_4636411027072998995t_unit
= ( ^ [G3: partia4934656038542163276t_unit,X4: set_int,A5: set_int,B7: set_int] :
( ( factor5186451337065598620t_unit @ G3 @ X4 @ A5 )
& ( factor5186451337065598620t_unit @ G3 @ X4 @ B7 )
& ! [Y6: set_int] :
( ( member_set_int @ Y6 @ ( partia966996272515721803t_unit @ G3 ) )
=> ( ( ( factor5186451337065598620t_unit @ G3 @ Y6 @ A5 )
& ( factor5186451337065598620t_unit @ G3 @ Y6 @ B7 ) )
=> ( factor5186451337065598620t_unit @ G3 @ Y6 @ X4 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_1015_isgcd__def,axiom,
( isgcd_1050045102061791863t_unit
= ( ^ [G3: partia4692342223508353374t_unit,X4: nat,A5: nat,B7: nat] :
( ( factor7017787129603596992t_unit @ G3 @ X4 @ A5 )
& ( factor7017787129603596992t_unit @ G3 @ X4 @ B7 )
& ! [Y6: nat] :
( ( member_nat @ Y6 @ ( partia3499330772048238685t_unit @ G3 ) )
=> ( ( ( factor7017787129603596992t_unit @ G3 @ Y6 @ A5 )
& ( factor7017787129603596992t_unit @ G3 @ Y6 @ B7 ) )
=> ( factor7017787129603596992t_unit @ G3 @ Y6 @ X4 ) ) ) ) ) ) ).
% isgcd_def
thf(fact_1016_ring_Osubfield__m__inv__simprule,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int,K2: set_int,A: set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( subfie3888952257595785920t_unit @ K @ R3 )
=> ( ( member_set_int @ K2 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ bot_bot_set_set_int ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ ( mult_s3864001451298473021t_unit @ R3 @ K2 @ A ) @ K )
=> ( member_set_int @ A @ K ) ) ) ) ) ) ).
% ring.subfield_m_inv_simprule
thf(fact_1017_ring_Osubfield__m__inv__simprule,axiom,
! [R3: partia4692342223508353374t_unit,K: set_nat,K2: nat,A: nat] :
( ( ring_n9194430563101542159t_unit @ R3 )
=> ( ( subfie4892355163478727762t_unit @ K @ R3 )
=> ( ( member_nat @ K2 @ ( minus_minus_set_nat @ K @ ( insert_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ bot_bot_set_nat ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ ( mult_n6028127365542633569t_unit @ R3 @ K2 @ A ) @ K )
=> ( member_nat @ A @ K ) ) ) ) ) ) ).
% ring.subfield_m_inv_simprule
thf(fact_1018_subfieldI,axiom,
! [K: set_set_int] :
( ( subcri1024317279029940167t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ( units_4038138251425117394t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : K
@ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
=> ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% subfieldI
thf(fact_1019_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1020_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1021_Units__closed,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% Units_closed
thf(fact_1022_Units__assoc,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ A @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ) ).
% Units_assoc
thf(fact_1023_unit__factor,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ A @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% unit_factor
thf(fact_1024_prod__unit__r,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ A @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_1025_prod__unit__l,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ B @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_1026_Units__cong,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ A @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ B @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% Units_cong
thf(fact_1027_unit__divides,axiom,
! [U: set_int,A: set_int] :
( ( member_set_int @ U @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ U @ A ) ) ) ).
% unit_divides
thf(fact_1028_divides__unit,axiom,
! [A: set_int,U: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ U )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ U @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ A @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% divides_unit
thf(fact_1029_Units__inv__comm,axiom,
! [X2: set_int,Y: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X2 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% Units_inv_comm
thf(fact_1030_ideal__eq__carrier__iff,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A ) )
= ( member_set_int @ A @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% ideal_eq_carrier_iff
thf(fact_1031_properfactor__unitE,axiom,
! [U: set_int,A: set_int] :
( ( member_set_int @ U @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( proper1002977052347345036t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ U )
=> ~ ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% properfactor_unitE
thf(fact_1032_add_Oint__pow__diff,axiom,
! [X2: set_int,N: int,M: int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( minus_minus_int @ N @ M ) @ X2 )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ X2 ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ X2 ) ) ) ) ) ).
% add.int_pow_diff
thf(fact_1033_Units__r__inv__ex,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ X )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% Units_r_inv_ex
thf(fact_1034_Units__l__inv__ex,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ X2 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% Units_l_inv_ex
thf(fact_1035_associatedI2_H,axiom,
! [A: set_int,B: set_int,U: set_int] :
( ( A
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ U ) )
=> ( ( member_set_int @ U @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ) ) ).
% associatedI2'
thf(fact_1036_associatedI2,axiom,
! [U: set_int,A: set_int,B: set_int] :
( ( member_set_int @ U @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( A
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ U ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( associ3816808824722140549t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ) ) ).
% associatedI2
thf(fact_1037_divides__one,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( member_set_int @ A @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% divides_one
thf(fact_1038_Unit__eq__dividesone,axiom,
! [U: set_int] :
( ( member_set_int @ U @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ U @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ U @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% Unit_eq_dividesone
thf(fact_1039_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1040_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1041_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1042_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1043_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1044_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_1045_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_1046_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_1047_Units__m__closed,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y ) @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% Units_m_closed
thf(fact_1048_Units__one__closed,axiom,
member_set_int @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% Units_one_closed
thf(fact_1049_Units__l__cancel,axiom,
! [X2: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X2 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% Units_l_cancel
thf(fact_1050_Units__minus__one__closed,axiom,
member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% Units_minus_one_closed
thf(fact_1051_diff__commute,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K2 ) @ J2 ) ) ).
% diff_commute
thf(fact_1052_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1053_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_1054_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1055_less__imp__diff__less,axiom,
! [J2: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J2 @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_1056_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_1057_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_1058_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1059_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_1060_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1061_le__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1062_eq__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1063_minus__set__def,axiom,
( minus_3247115583872269408et_int
= ( ^ [A4: set_nat_set_int,B4: set_nat_set_int] :
( collect_nat_set_int
@ ( minus_3277206198080951333_int_o
@ ^ [X4: nat > set_int] : ( member_nat_set_int @ X4 @ A4 )
@ ^ [X4: nat > set_int] : ( member_nat_set_int @ X4 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1064_minus__set__def,axiom,
( minus_5256904771846099296nt_nat
= ( ^ [A4: set_set_int_nat,B4: set_set_int_nat] :
( collect_set_int_nat
@ ( minus_5750293070438678309_nat_o
@ ^ [X4: set_int > nat] : ( member_set_int_nat @ X4 @ A4 )
@ ^ [X4: set_int > nat] : ( member_set_int_nat @ X4 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1065_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1066_minus__set__def,axiom,
( minus_6857623457997529383et_int
= ( ^ [A4: set_set_set_int,B4: set_set_set_int] :
( collect_set_set_int
@ ( minus_535336280041981470_int_o
@ ^ [X4: set_set_int] : ( member_set_set_int @ X4 @ A4 )
@ ^ [X4: set_set_int] : ( member_set_set_int @ X4 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1067_minus__set__def,axiom,
( minus_8897228262479074673et_int
= ( ^ [A4: set_set_int,B4: set_set_int] :
( collect_set_int
@ ( minus_4665002624458620500_int_o
@ ^ [X4: set_int] : ( member_set_int @ X4 @ A4 )
@ ^ [X4: set_int] : ( member_set_int @ X4 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_1068_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_1069_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1070_less__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1071_ring_OUnits__minus__one__closed,axiom,
! [R3: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ R3 @ ( one_se8065767436706823081t_unit @ R3 ) ) @ ( units_4038138251425117394t_unit @ R3 ) ) ) ).
% ring.Units_minus_one_closed
thf(fact_1072_cring_Oideal__eq__carrier__iff,axiom,
! [R3: partia4934656038542163276t_unit,A: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( ( partia966996272515721803t_unit @ R3 )
= ( cgenid8502489213727343375t_unit @ R3 @ A ) )
= ( member_set_int @ A @ ( units_4038138251425117394t_unit @ R3 ) ) ) ) ) ).
% cring.ideal_eq_carrier_iff
thf(fact_1073_cring_Oideal__eq__carrier__iff,axiom,
! [R3: partia4692342223508353374t_unit,A: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( ( partia3499330772048238685t_unit @ R3 )
= ( cgenid8693976350862946099t_unit @ R3 @ A ) )
= ( member_nat @ A @ ( units_1295200668666280182t_unit @ R3 ) ) ) ) ) ).
% cring.ideal_eq_carrier_iff
thf(fact_1074_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( monoid497721730651901107t_unit @ G )
=> ( ( member_set_int @ A @ ( units_4038138251425117394t_unit @ G ) )
=> ( ( associ3816808824722140549t_unit @ G @ A @ B )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G ) )
=> ( member_set_int @ B @ ( units_4038138251425117394t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_1075_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( monoid2713784563552164055t_unit @ G )
=> ( ( member_nat @ A @ ( units_1295200668666280182t_unit @ G ) )
=> ( ( associ4357985397585971625t_unit @ G @ A @ B )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G ) )
=> ( member_nat @ B @ ( units_1295200668666280182t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_1076_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( monoid497721730651901107t_unit @ G )
=> ( ( associ3816808824722140549t_unit @ G @ A @ B )
=> ( ( member_set_int @ B @ ( units_4038138251425117394t_unit @ G ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G ) )
=> ( member_set_int @ A @ ( units_4038138251425117394t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_1077_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( monoid2713784563552164055t_unit @ G )
=> ( ( associ4357985397585971625t_unit @ G @ A @ B )
=> ( ( member_nat @ B @ ( units_1295200668666280182t_unit @ G ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G ) )
=> ( member_nat @ A @ ( units_1295200668666280182t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_1078_cring_Odivides__one,axiom,
! [R3: partia4934656038542163276t_unit,A: set_int] :
( ( cring_3079150759069666002t_unit @ R3 )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( factor5186451337065598620t_unit @ R3 @ A @ ( one_se8065767436706823081t_unit @ R3 ) )
= ( member_set_int @ A @ ( units_4038138251425117394t_unit @ R3 ) ) ) ) ) ).
% cring.divides_one
thf(fact_1079_cring_Odivides__one,axiom,
! [R3: partia4692342223508353374t_unit,A: nat] :
( ( cring_4736503688146807908t_unit @ R3 )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( factor7017787129603596992t_unit @ R3 @ A @ ( one_na902338870878123981t_unit @ R3 ) )
= ( member_nat @ A @ ( units_1295200668666280182t_unit @ R3 ) ) ) ) ) ).
% cring.divides_one
thf(fact_1080_monoid__cancel_Oassociated__iff,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( monoid497721730651901107t_unit @ G )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G ) )
=> ( ( associ3816808824722140549t_unit @ G @ A @ B )
= ( ? [X4: set_int] :
( ( member_set_int @ X4 @ ( units_4038138251425117394t_unit @ G ) )
& ( A
= ( mult_s3864001451298473021t_unit @ G @ B @ X4 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_1081_monoid__cancel_Oassociated__iff,axiom,
! [G: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( monoid2713784563552164055t_unit @ G )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( associ4357985397585971625t_unit @ G @ A @ B )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( units_1295200668666280182t_unit @ G ) )
& ( A
= ( mult_n6028127365542633569t_unit @ G @ B @ X4 ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_1082_monoid__cancel_OassociatedE2,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( monoid497721730651901107t_unit @ G )
=> ( ( associ3816808824722140549t_unit @ G @ A @ B )
=> ( ! [U3: set_int] :
( ( A
= ( mult_s3864001451298473021t_unit @ G @ B @ U3 ) )
=> ~ ( member_set_int @ U3 @ ( units_4038138251425117394t_unit @ G ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G ) )
=> ~ ( member_set_int @ B @ ( partia966996272515721803t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_1083_monoid__cancel_OassociatedE2,axiom,
! [G: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( monoid2713784563552164055t_unit @ G )
=> ( ( associ4357985397585971625t_unit @ G @ A @ B )
=> ( ! [U3: nat] :
( ( A
= ( mult_n6028127365542633569t_unit @ G @ B @ U3 ) )
=> ~ ( member_nat @ U3 @ ( units_1295200668666280182t_unit @ G ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G ) )
=> ~ ( member_nat @ B @ ( partia3499330772048238685t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_1084_monoid__cancel_OassociatedD2,axiom,
! [G: partia4934656038542163276t_unit,A: set_int,B: set_int] :
( ( monoid497721730651901107t_unit @ G )
=> ( ( associ3816808824722140549t_unit @ G @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ G ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ G ) )
=> ? [X: set_int] :
( ( member_set_int @ X @ ( units_4038138251425117394t_unit @ G ) )
& ( A
= ( mult_s3864001451298473021t_unit @ G @ B @ X ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_1085_monoid__cancel_OassociatedD2,axiom,
! [G: partia4692342223508353374t_unit,A: nat,B: nat] :
( ( monoid2713784563552164055t_unit @ G )
=> ( ( associ4357985397585971625t_unit @ G @ A @ B )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ G ) )
=> ( ( member_nat @ B @ ( partia3499330772048238685t_unit @ G ) )
=> ? [X: nat] :
( ( member_nat @ X @ ( units_1295200668666280182t_unit @ G ) )
& ( A
= ( mult_n6028127365542633569t_unit @ G @ B @ X ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_1086_subfield_Osubfield__Units,axiom,
! [K: set_set_int,R3: partia4934656038542163276t_unit] :
( ( subfie3888952257595785920t_unit @ K @ R3 )
=> ( ( units_4038138251425117394t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : K
@ R3 ) )
= ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ bot_bot_set_set_int ) ) ) ) ).
% subfield.subfield_Units
thf(fact_1087_ring_OsubfieldI,axiom,
! [R3: partia4934656038542163276t_unit,K: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R3 )
=> ( ( subcri1024317279029940167t_unit @ K @ R3 )
=> ( ( ( units_4038138251425117394t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : K
@ R3 ) )
= ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ bot_bot_set_set_int ) ) )
=> ( subfie3888952257595785920t_unit @ K @ R3 ) ) ) ) ).
% ring.subfieldI
thf(fact_1088_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_1089_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_1090_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_1091_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1092_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1093_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_1094_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_1095_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_1096_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_1097_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_1098_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1099_field__intro2,axiom,
( ( ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ! [X: set_int] :
( ( member_set_int @ X @ ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
=> ( member_set_int @ X @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
=> ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% field_intro2
thf(fact_1100_cring__fieldI,axiom,
( ( ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
=> ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% cring_fieldI
thf(fact_1101_subfield__m__inv_I2_J,axiom,
! [K: set_set_int,K2: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ K2 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 ) )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% subfield_m_inv(2)
thf(fact_1102_subfield__m__inv_I3_J,axiom,
! [K: set_set_int,K2: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ K2 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 ) @ K2 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% subfield_m_inv(3)
thf(fact_1103_inv__eq__imp__eq,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 )
= ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) )
=> ( X2 = Y ) ) ) ) ).
% inv_eq_imp_eq
thf(fact_1104_inv__eq__one__eq,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( X2
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% inv_eq_one_eq
thf(fact_1105_subfield__iff_I2_J,axiom,
! [K: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( field_5943785737635511755t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : K
@ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% subfield_iff(2)
thf(fact_1106_comm__inv__char,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 )
= Y ) ) ) ) ).
% comm_inv_char
thf(fact_1107_inv__char,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 )
= Y ) ) ) ) ) ).
% inv_char
thf(fact_1108_inv__unique_H,axiom,
! [X2: set_int,Y: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ Y )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X2 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( Y
= ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) ) ) ) ) ) ).
% inv_unique'
thf(fact_1109_inv__eq__neg__one__eq,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
= ( X2
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% inv_eq_neg_one_eq
thf(fact_1110_cring__fieldI2,axiom,
( ( ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ! [A3: set_int] :
( ( member_set_int @ A3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( A3
!= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [X6: set_int] :
( ( member_set_int @ X6 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A3 @ X6 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) )
=> ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% cring_fieldI2
thf(fact_1111_subfield__iff_I1_J,axiom,
! [K: set_set_int] :
( ( field_5943785737635511755t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : K
@ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% subfield_iff(1)
thf(fact_1112_subfield__m__inv_I1_J,axiom,
! [K: set_set_int,K2: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ K2 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
=> ( member_set_int @ ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 ) @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) ) ) ) ).
% subfield_m_inv(1)
thf(fact_1113_inv__one,axiom,
( ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% inv_one
thf(fact_1114_Units__inv__Units,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% Units_inv_Units
thf(fact_1115_Units__inv__inv,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) )
= X2 ) ) ).
% Units_inv_inv
thf(fact_1116_Units__inv__closed,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% Units_inv_closed
thf(fact_1117_inv__neg__one,axiom,
( ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% inv_neg_one
thf(fact_1118_Units__l__inv,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ X2 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% Units_l_inv
thf(fact_1119_Units__r__inv,axiom,
! [X2: set_int] :
( ( member_set_int @ X2 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% Units_r_inv
thf(fact_1120_zeromaximalideal__eq__field,axiom,
( ( maxima6262477034536100350t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% zeromaximalideal_eq_field
thf(fact_1121_zeromaximalideal__fieldI,axiom,
( ( maxima6262477034536100350t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% zeromaximalideal_fieldI
thf(fact_1122_m__inv__monoid__consistent,axiom,
! [X2: set_int,H3: set_set_int] :
( ( member_set_int @ X2
@ ( units_4038138251425117394t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : H3
@ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
=> ( ( submon6016771835709735619t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( m_inv_4894562657074299959t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : H3
@ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
@ X2 )
= ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) ) ) ) ).
% m_inv_monoid_consistent
thf(fact_1123_maximalideal__prime,axiom,
! [I3: set_set_int] :
( ( maxima6262477034536100350t_unit @ I3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( primei350866878734230858t_unit @ I3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% maximalideal_prime
thf(fact_1124_space__subgroup__props_I6_J,axiom,
! [K: set_set_int,N: nat,E: set_set_int,K2: set_int,A: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( ( member_set_int @ K2 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ A ) @ E )
=> ( member_set_int @ A @ E ) ) ) ) ) ) ).
% space_subgroup_props(6)
thf(fact_1125_subringI,axiom,
! [H3: set_set_int] :
( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ H3 )
=> ( ! [H4: set_int] :
( ( member_set_int @ H4 @ H3 )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H4 ) @ H3 ) )
=> ( ! [H12: set_int,H22: set_int] :
( ( member_set_int @ H12 @ H3 )
=> ( ( member_set_int @ H22 @ H3 )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H12 @ H22 ) @ H3 ) ) )
=> ( ! [H12: set_int,H22: set_int] :
( ( member_set_int @ H12 @ H3 )
=> ( ( member_set_int @ H22 @ H3 )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H12 @ H22 ) @ H3 ) ) )
=> ( subrin7689096310803670856t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ) ).
% subringI
thf(fact_1126_carrier__is__subring,axiom,
subrin7689096310803670856t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% carrier_is_subring
thf(fact_1127_dimension__is__inj,axiom,
! [K: set_set_int,N: nat,E: set_set_int,M: nat] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ K @ E )
=> ( N = M ) ) ) ) ).
% dimension_is_inj
thf(fact_1128_finite__dimensionE_H,axiom,
! [K: set_set_int,E: set_set_int] :
( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ E )
=> ~ ! [N2: nat] :
~ ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ K @ E ) ) ).
% finite_dimensionE'
thf(fact_1129_finite__dimensionI,axiom,
! [N: nat,K: set_set_int,E: set_set_int] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ E ) ) ).
% finite_dimensionI
thf(fact_1130_finite__dimension__def,axiom,
! [K: set_set_int,E: set_set_int] :
( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ E )
= ( ? [N4: nat] : ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N4 @ K @ E ) ) ) ).
% finite_dimension_def
thf(fact_1131_subcringI_H,axiom,
! [H3: set_set_int] :
( ( subrin7689096310803670856t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( subcri1024317279029940167t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% subcringI'
thf(fact_1132_space__subgroup__props_I2_J,axiom,
! [K: set_set_int,N: nat,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ E ) ) ) ).
% space_subgroup_props(2)
thf(fact_1133_space__subgroup__props_I3_J,axiom,
! [K: set_set_int,N: nat,E: set_set_int,V1: set_int,V22: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( ( member_set_int @ V1 @ E )
=> ( ( member_set_int @ V22 @ E )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ V1 @ V22 ) @ E ) ) ) ) ) ).
% space_subgroup_props(3)
thf(fact_1134_telescopic__base__aux,axiom,
! [K: set_set_int,F3: set_set_int,N: nat,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( subfie3888952257595785920t_unit @ F3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ F3 )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ one_one_nat @ F3 @ E )
=> ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E ) ) ) ) ) ).
% telescopic_base_aux
thf(fact_1135_space__subgroup__props_I5_J,axiom,
! [K: set_set_int,N: nat,E: set_set_int,K2: set_int,V4: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( ( member_set_int @ K2 @ K )
=> ( ( member_set_int @ V4 @ E )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ V4 ) @ E ) ) ) ) ) ).
% space_subgroup_props(5)
thf(fact_1136_space__subgroup__props_I4_J,axiom,
! [K: set_set_int,N: nat,E: set_set_int,V4: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( ( member_set_int @ V4 @ E )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ V4 ) @ E ) ) ) ) ).
% space_subgroup_props(4)
thf(fact_1137_unique__dimension,axiom,
! [K: set_set_int,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ E )
=> ? [X: nat] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ K @ E )
& ! [Y4: nat] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y4 @ K @ E )
=> ( Y4 = X ) ) ) ) ) ).
% unique_dimension
thf(fact_1138_subcringI,axiom,
! [H3: set_set_int] :
( ( subrin7689096310803670856t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ! [H12: set_int,H22: set_int] :
( ( member_set_int @ H12 @ H3 )
=> ( ( member_set_int @ H22 @ H3 )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H12 @ H22 )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H22 @ H12 ) ) ) )
=> ( subcri1024317279029940167t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% subcringI
thf(fact_1139_add__pow__consistent,axiom,
! [K: set_set_int,K2: set_int,I2: int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ K2 @ K )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ K2 )
= ( add_po7581009264371422883it_int
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : K
@ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
@ I2
@ K2 ) ) ) ) ).
% add_pow_consistent
thf(fact_1140_line__extension__consistent,axiom,
! [K: set_set_int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd4283282269743769663t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : K
@ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% line_extension_consistent
thf(fact_1141_space__subgroup__props_I1_J,axiom,
! [K: set_set_int,N: nat,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% space_subgroup_props(1)
thf(fact_1142_subring__is__ring,axiom,
! [H3: set_set_int] :
( ( subrin7689096310803670856t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ring_s5316885176909347197t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : H3
@ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% subring_is_ring
thf(fact_1143_zero__dim,axiom,
! [K: set_set_int] : ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ zero_zero_nat @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) ).
% zero_dim
thf(fact_1144_dimension__zero,axiom,
! [K: set_set_int,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ zero_zero_nat @ K @ E )
=> ( E
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) ) ) ).
% dimension_zero
thf(fact_1145_ring__incl__imp__subring,axiom,
! [H3: set_set_int] :
( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ring_s5316885176909347197t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : H3
@ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( subrin7689096310803670856t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% ring_incl_imp_subring
thf(fact_1146_subring__iff,axiom,
! [H3: set_set_int] :
( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( subrin7689096310803670856t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( ring_s5316885176909347197t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : H3
@ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% subring_iff
thf(fact_1147_dimension__one,axiom,
! [K: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ one_one_nat @ K @ K ) ) ).
% dimension_one
thf(fact_1148_poly__of__const__consistent,axiom,
! [K: set_set_int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( poly_o1995067004181805532t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : K
@ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( poly_o1995067004181805532t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% poly_of_const_consistent
thf(fact_1149_dimension_Ocases,axiom,
! [A1: nat,A22: set_set_int,A32: set_set_int] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
=> ~ ! [V2: set_int,E2: set_set_int,N2: nat] :
( ( A1
= ( suc @ N2 ) )
=> ( ( A32
= ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A22 @ V2 @ E2 ) )
=> ( ( member_set_int @ V2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ~ ( member_set_int @ V2 @ E2 )
=> ~ ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ A22 @ E2 ) ) ) ) ) ) ) ).
% dimension.cases
thf(fact_1150_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1151_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1152_Suc__dim,axiom,
! [V4: set_int,E: set_set_int,N: nat,K: set_set_int] :
( ( member_set_int @ V4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ~ ( member_set_int @ V4 @ E )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E )
=> ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( suc @ N ) @ K @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ V4 @ E ) ) ) ) ) ).
% Suc_dim
thf(fact_1153_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1154_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1155_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_1156_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_1157_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_1158_Suc__diff__diff,axiom,
! [M: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_1159_dimension__backwards,axiom,
! [K: set_set_int,N: nat,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( suc @ N ) @ K @ E )
=> ? [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ? [E3: set_set_int] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K @ E3 )
& ~ ( member_set_int @ X @ E3 )
& ( E
= ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ X @ E3 ) ) ) ) ) ) ).
% dimension_backwards
thf(fact_1160_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1161_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1162_dimension_Osimps,axiom,
! [A1: nat,A22: set_set_int,A32: set_set_int] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A1 @ A22 @ A32 )
= ( ? [K4: set_set_int] :
( ( A1 = zero_zero_nat )
& ( A22 = K4 )
& ( A32
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
| ? [V5: set_int,E4: set_set_int,N4: nat,K4: set_set_int] :
( ( A1
= ( suc @ N4 ) )
& ( A22 = K4 )
& ( A32
= ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K4 @ V5 @ E4 ) )
& ( member_set_int @ V5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ~ ( member_set_int @ V5 @ E4 )
& ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N4 @ K4 @ E4 ) ) ) ) ).
% dimension.simps
thf(fact_1163_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1164_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_1165_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1166_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I2: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1167_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1168_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1169_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1170_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1171_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1172_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1173_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X: nat] : ( P @ X @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X: nat,Y3: nat] :
( ( P @ X @ Y3 )
=> ( P @ ( suc @ X ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1174_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1175_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1176_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1177_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1178_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_1179_Nat_OlessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( ( K2
!= ( suc @ I2 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1180_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1181_Suc__lessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K2 )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_1182_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1183_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1184_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1185_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_1186_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1187_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1188_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_1189_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1190_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1191_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1192_less__trans__Suc,axiom,
! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ord_less_nat @ ( suc @ I2 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_1193_less__Suc__induct,axiom,
! [I2: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P @ I4 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I4 @ K3 ) ) ) ) )
=> ( P @ I2 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_1194_strict__inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ! [I4: nat] :
( ( J2
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_1195_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1196_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X: nat] : ( R3 @ X @ X )
=> ( ! [X: nat,Y3: nat,Z2: nat] :
( ( R3 @ X @ Y3 )
=> ( ( R3 @ Y3 @ Z2 )
=> ( R3 @ X @ Z2 ) ) )
=> ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
=> ( R3 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1197_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1198_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1199_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_1200_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1201_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_1202_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M3: nat] :
( M7
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_1203_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1204_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_1205_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1206_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1207_Suc__inject,axiom,
! [X2: nat,Y: nat] :
( ( ( suc @ X2 )
= ( suc @ Y ) )
=> ( X2 = Y ) ) ).
% Suc_inject
thf(fact_1208_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J4: nat] :
( ( M
= ( suc @ J4 ) )
& ( ord_less_nat @ J4 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1209_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1210_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1211_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1212_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1213_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1214_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1215_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1216_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1217_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1218_inc__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1219_dec__induct,axiom,
! [I2: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( P @ I2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_1220_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1221_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1222_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1223_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1224_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1225_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_1226_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1227_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1228_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1229_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1230_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1231_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1232_char__consistent,axiom,
! [H3: set_set_int] :
( ( subrin7689096310803670856t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ring_c6147214092195050492t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : H3
@ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% char_consistent
thf(fact_1233_subdomain__iff,axiom,
! [H3: set_set_int] :
( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( subdom1520866149873910708t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( domain6183376680155302761t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : H3
@ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% subdomain_iff
thf(fact_1234_subdomain__is__domain,axiom,
! [H3: set_set_int] :
( ( subdom1520866149873910708t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( domain6183376680155302761t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : H3
@ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% subdomain_is_domain
thf(fact_1235_domain__eq__zeroprimeideal,axiom,
( ( domain6183376680155302761t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( primei350866878734230858t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% domain_eq_zeroprimeideal
thf(fact_1236_zeroprimeideal__domainI,axiom,
( ( primei350866878734230858t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( domain6183376680155302761t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% zeroprimeideal_domainI
thf(fact_1237_char__bound_I1_J,axiom,
! [X2: nat] :
( ( ord_less_nat @ zero_zero_nat @ X2 )
=> ( ( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( semiri1314217659103216013at_int @ X2 ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_less_eq_nat @ ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X2 ) ) ) ).
% char_bound(1)
thf(fact_1238_finite__carr__imp__char__ge__0,axiom,
( ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% finite_carr_imp_char_ge_0
thf(fact_1239_int__embed__closed,axiom,
! [K2: int] : ( member_set_int @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% int_embed_closed
thf(fact_1240_int__embed__zero,axiom,
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ zero_zero_int )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% int_embed_zero
thf(fact_1241_int__embed__range,axiom,
! [K: set_set_int,K2: int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( member_set_int @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 ) @ K ) ) ).
% int_embed_range
thf(fact_1242_int__embed__one,axiom,
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ one_one_int )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% int_embed_one
thf(fact_1243_embed__char__eq__0,axiom,
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( semiri1314217659103216013at_int @ ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% embed_char_eq_0
thf(fact_1244_int__embed__diff,axiom,
! [X2: int,Y: int] :
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( minus_minus_int @ X2 @ Y ) )
= ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ).
% int_embed_diff
thf(fact_1245_int__embed__consistent,axiom,
! [K: set_set_int,I2: int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 )
= ( ring_i2743490682209504680t_unit
@ ( partia2914870419229397556t_unit
@ ^ [Uu: set_set_int] : K
@ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
@ I2 ) ) ) ).
% int_embed_consistent
thf(fact_1246_char__bound_I2_J,axiom,
! [X2: nat] :
( ( ord_less_nat @ zero_zero_nat @ X2 )
=> ( ( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( semiri1314217659103216013at_int @ X2 ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% char_bound(2)
thf(fact_1247_finite__ring__finite__units,axiom,
( ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( finite6197958912794628473et_int @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% finite_ring_finite_units
thf(fact_1248_order__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_4716970363388151434t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% order_gt_0_iff_finite
thf(fact_1249_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_1250_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_1251_mod__ring__finite,axiom,
! [N: nat] : ( finite_finite_nat @ ( partia3499330772048238685t_unit @ ( mod_ring @ N ) ) ) ).
% mod_ring_finite
thf(fact_1252_fin__zfact,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% fin_zfact
thf(fact_1253_zdiff__int__split,axiom,
! [P: int > $o,X2: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X2 @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1254_rcosets__subset__PowG,axiom,
! [H3: set_set_int] :
( ( additi7073586575563672860t_unit @ H3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ord_le4317611570275147438et_int @ ( a_RCOS5559887075240879033t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H3 ) @ ( pow_set_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% rcosets_subset_PowG
thf(fact_1255_conj__le__cong,axiom,
! [X2: int,X3: int,P: $o,P3: $o] :
( ( X2 = X3 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X3 )
& P3 ) ) ) ) ).
% conj_le_cong
thf(fact_1256_imp__le__cong,axiom,
! [X2: int,X3: int,P: $o,P3: $o] :
( ( X2 = X3 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> P3 ) ) ) ) ).
% imp_le_cong
thf(fact_1257_a__card__cosets__equal,axiom,
! [C: set_set_int,H3: set_set_int] :
( ( member_set_set_int @ C @ ( a_RCOS5559887075240879033t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H3 ) )
=> ( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( finite_card_set_int @ C )
= ( finite_card_set_int @ H3 ) ) ) ) ) ).
% a_card_cosets_equal
thf(fact_1258_trivialideals__eq__field,axiom,
( ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
=> ( ( ( collect_set_set_int
@ ^ [I6: set_set_int] : ( ideal_7262958097527394045t_unit @ I6 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( insert_set_set_int @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ ( insert_set_set_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bo2384636101374064866et_int ) ) )
= ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% trivialideals_eq_field
thf(fact_1259_oneideal,axiom,
ideal_7262958097527394045t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% oneideal
thf(fact_1260_a__rcos__zero,axiom,
! [I3: set_set_int,I2: set_int] :
( ( ideal_7262958097527394045t_unit @ I3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ I2 @ I3 )
=> ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I3 @ I2 )
= I3 ) ) ) ).
% a_rcos_zero
thf(fact_1261_add__ideals,axiom,
! [I3: set_set_int,J: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ideal_7262958097527394045t_unit @ J @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ideal_7262958097527394045t_unit @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I3 @ J ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% add_ideals
thf(fact_1262_cgenideal__ideal,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ideal_7262958097527394045t_unit @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% cgenideal_ideal
thf(fact_1263_genideal__minimal,axiom,
! [I3: set_set_int,S: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ord_le4403425263959731960et_int @ S @ I3 )
=> ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S ) @ I3 ) ) ) ).
% genideal_minimal
thf(fact_1264_cgenideal__minimal,axiom,
! [J: set_set_int,A: set_int] :
( ( ideal_7262958097527394045t_unit @ J @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ A @ J )
=> ( ord_le4403425263959731960et_int @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A ) @ J ) ) ) ).
% cgenideal_minimal
thf(fact_1265_zeroideal,axiom,
ideal_7262958097527394045t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% zeroideal
thf(fact_1266_genideal__ideal,axiom,
! [S: set_set_int] :
( ( ord_le4403425263959731960et_int @ S @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ideal_7262958097527394045t_unit @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% genideal_ideal
thf(fact_1267_Idl__subset__ideal,axiom,
! [I3: set_set_int,H3: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ord_le4403425263959731960et_int @ H3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H3 ) @ I3 )
= ( ord_le4403425263959731960et_int @ H3 @ I3 ) ) ) ) ).
% Idl_subset_ideal
thf(fact_1268_ideal__is__subalgebra,axiom,
! [K: set_set_int,I3: set_set_int] :
( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ideal_7262958097527394045t_unit @ I3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( embedd2743979684206749024t_unit @ K @ I3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% ideal_is_subalgebra
thf(fact_1269_quotient__eq__iff__same__a__r__cos,axiom,
! [I3: set_set_int,A: set_int,B: set_int] :
( ( ideal_7262958097527394045t_unit @ I3 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) @ I3 )
= ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I3 @ A )
= ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I3 @ B ) ) ) ) ) ) ).
% quotient_eq_iff_same_a_r_cos
thf(fact_1270_trivialideals__fieldI,axiom,
( ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
=> ( ( ( collect_set_set_int
@ ^ [I6: set_set_int] : ( ideal_7262958097527394045t_unit @ I6 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( insert_set_set_int @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ ( insert_set_set_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bo2384636101374064866et_int ) ) )
=> ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% trivialideals_fieldI
% Helper facts (3)
thf(help_If_3_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X2: set_int,Y: set_int] :
( ( if_set_int @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
! [X2: set_int,Y: set_int] :
( ( if_set_int @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( zero_u3727541067695463886t_unit
@ ^ [Uu: nat] : ( zfact_iso_inv @ n @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
@ ( mod_ring @ n ) )
= ( mod_ring @ n ) ) ).
%------------------------------------------------------------------------------