TPTP Problem File: SLH0645^1.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_00094_002984__18250292_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1403 ( 584 unt; 128 typ; 0 def)
% Number of atoms : 3460 (1331 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 13021 ( 300 ~; 64 |; 190 &;11095 @)
% ( 0 <=>;1372 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 375 ( 375 >; 0 *; 0 +; 0 <<)
% Number of symbols : 118 ( 115 usr; 12 con; 0-4 aty)
% Number of variables : 3097 ( 79 ^;2962 !; 56 ?;3097 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:40:43.780
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
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,
partia4934656038542163276t_unit: $tType ).
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,
partia4692342223508353374t_unit: $tType ).
thf(ty_n_t__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,type,
monoid6410947536436496832t_unit: $tType ).
thf(ty_n_t__Group__Omonoid__Omonoid____ext_It__Nat__Onat_Mt__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
monoid5117177112048413924t_unit: $tType ).
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,
set_set_int_set_int: $tType ).
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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_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 (115)
thf(sy_c_AbelCoset_Oa__l__coset_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
a_l_co3504123944629134560t_unit: partia4934656038542163276t_unit > set_int > set_set_int > set_set_int ).
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__Set__Oset_It__Int__Oint_J_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,
partia3499330772048238685t_unit: partia4692342223508353374t_unit > set_nat ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Set__Oset_It__Int__Oint_J_001t__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,type,
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thf(sy_c_Congruence_Opartial__object_Ocarrier__update_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_Congruence_Opartial__object_Ocarrier__update_001t__Set__Oset_It__Int__Oint_J_001t__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,type,
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thf(sy_c_Congruence_Opartial__object_Omore_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_Congruence_Opartial__object_Omore_001t__Set__Oset_It__Int__Oint_J_001t__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,type,
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thf(sy_c_Congruence_Opartial__object_Opartial__object__ext_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_Congruence_Opartial__object_Opartial__object__ext_001t__Set__Oset_It__Int__Oint_J_001t__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,type,
partia768359127423289238t_unit: set_set_int > monoid6410947536436496832t_unit > partia4934656038542163276t_unit ).
thf(sy_c_Coset_Oset__mult_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,
set_mu2785919024023201382t_unit: partia4934656038542163276t_unit > set_set_int > set_set_int > set_set_int ).
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_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__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Embedded__Algebras_Osubalgebra_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
embedd2743979684206749024t_unit: set_set_int > set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Field_Omod__ring,type,
mod_ring: nat > partia4692342223508353374t_unit ).
thf(sy_c_Field_Ozfact__iso__inv,type,
zfact_iso_inv: nat > set_int > nat ).
thf(sy_c_Group_OUnits_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_Group_Om__inv_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_Group_Omonoid_Omult_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__Set__Oset_It__Int__Oint_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J,type,
mult_s3864001451298473021t_unit: partia4934656038542163276t_unit > set_int > set_int > set_int ).
thf(sy_c_Group_Omonoid_Oone_001t__Nat__Onat_001t__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
one_na902338870878123981t_unit: partia4692342223508353374t_unit > nat ).
thf(sy_c_Group_Omonoid_Oone_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,
one_se8065767436706823081t_unit: partia4934656038542163276t_unit > set_int ).
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,
minus_3247115583872269408et_int: set_nat_set_int > set_nat_set_int > set_nat_set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
minus_8897228262479074673et_int: set_set_int > set_set_int > set_set_int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
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,
zero_zero_nat: nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
hilber3159750049796175616et_int: set_nat > ( nat > set_int ) > set_int > nat ).
thf(sy_c_Ideal_Ocgenideal_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_Ideal_Ogenideal_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
genide1545711809618862555t_unit: partia4934656038542163276t_unit > set_set_int > set_set_int ).
thf(sy_c_Ideal_Omaximalideal_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Ideal_Oprimeideal_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Ideal_Oprincipalideal_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
princi8860937869964495385t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_IntRing_OZFact,type,
zFact: int > partia4934656038542163276t_unit ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
inf_in1752217752563533465et_int: set_nat_set_int > set_nat_set_int > set_nat_set_int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
inf_inf_set_set_int: set_set_int > set_set_int > set_set_int ).
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_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Int__Oint_J_M_Eo_J,type,
bot_bot_set_int_o: set_int > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_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,
bot_bo8417611410066262939et_int: set_nat_set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
bot_bot_set_set_int: set_set_int ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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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,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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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__eq_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
ord_le5995675665013768039et_int: set_nat_set_int > set_nat_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_QuotRing_Omorphic__prop_001t__Nat__Onat_001t__Product____Type__Ounit,type,
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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,
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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,
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thf(sy_c_Ring_Oadd__pow_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Nat__Onat,type,
add_po7583499734880473159it_nat: partia4934656038542163276t_unit > nat > set_int > set_int ).
thf(sy_c_Ring_Ofield_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
field_5943785737635511755t_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_Osemiring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
semiri8708897239777792527t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring__Characteristic_Ozfact__iso,type,
ring_zfact_iso: nat > nat > set_int ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
collect_nat_set_int: ( ( nat > set_int ) > $o ) > set_nat_set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
collect_set_int: ( set_int > $o ) > 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_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__Interval_Oord__class_OlessThan_001_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
set_or7260056672446632558et_int: ( nat > set_int ) > set_nat_set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Int__Oint_J,type,
set_or5935648273017318783et_int: set_int > 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__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
subfie3888952257595785920t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_UnivPoly_OUP__univ__prop__axioms_001t__Nat__Onat_001t__Product____Type__Ounit,type,
uP_uni9161691548522291798t_unit: partia4692342223508353374t_unit > nat > $o ).
thf(sy_c_UnivPoly_OUP__univ__prop__axioms_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
uP_uni688348660276476100t_unit: partia4934656038542163276t_unit > set_int > $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_v_n,type,
n: nat ).
thf(sy_v_x____,type,
x: set_int ).
% Relevant facts (1271)
thf(fact_0__092_060open_062zfact__iso__inv_An_Ax_A_092_060in_062_A_123_O_O_060n_125_092_060close_062,axiom,
member_nat @ ( zfact_iso_inv @ n @ x ) @ ( set_ord_lessThan_nat @ n ) ).
% \<open>zfact_iso_inv n x \<in> {..<n}\<close>
thf(fact_1__092_060open_062x_A_092_060in_062_Acarrier_A_IZFact_A_Iint_An_J_J_092_060close_062,axiom,
member_set_int @ x @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% \<open>x \<in> carrier (ZFact (int n))\<close>
thf(fact_2_n__ge__1,axiom,
ord_less_nat @ one_one_nat @ n ).
% n_ge_1
thf(fact_3_n__ge__0,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% n_ge_0
thf(fact_4_UP__univ__prop__axioms_Ointro,axiom,
! [S: nat,S2: partia4692342223508353374t_unit] :
( ( member_nat @ S @ ( partia3499330772048238685t_unit @ S2 ) )
=> ( uP_uni9161691548522291798t_unit @ S2 @ S ) ) ).
% UP_univ_prop_axioms.intro
thf(fact_5_UP__univ__prop__axioms_Ointro,axiom,
! [S: set_int,S2: partia4934656038542163276t_unit] :
( ( member_set_int @ S @ ( partia966996272515721803t_unit @ S2 ) )
=> ( uP_uni688348660276476100t_unit @ S2 @ S ) ) ).
% UP_univ_prop_axioms.intro
thf(fact_6_UP__univ__prop__axioms__def,axiom,
( uP_uni9161691548522291798t_unit
= ( ^ [S3: partia4692342223508353374t_unit,S4: nat] : ( member_nat @ S4 @ ( partia3499330772048238685t_unit @ S3 ) ) ) ) ).
% UP_univ_prop_axioms_def
thf(fact_7_UP__univ__prop__axioms__def,axiom,
( uP_uni688348660276476100t_unit
= ( ^ [S3: partia4934656038542163276t_unit,S4: set_int] : ( member_set_int @ S4 @ ( partia966996272515721803t_unit @ S3 ) ) ) ) ).
% UP_univ_prop_axioms_def
thf(fact_8_partial__object_Ofold__congs_I1_J,axiom,
! [R: partia4692342223508353374t_unit,R2: partia4692342223508353374t_unit,V: set_nat,F: set_nat > set_nat,F2: set_nat > set_nat] :
( ( R = R2 )
=> ( ( ( partia3499330772048238685t_unit @ R2 )
= V )
=> ( ! [V2: set_nat] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( partia6063033193446304838t_unit @ F @ R )
= ( partia6063033193446304838t_unit @ F2 @ R2 ) ) ) ) ) ).
% partial_object.fold_congs(1)
thf(fact_9_partial__object_Ofold__congs_I1_J,axiom,
! [R: partia4934656038542163276t_unit,R2: partia4934656038542163276t_unit,V: set_set_int,F: set_set_int > set_set_int,F2: set_set_int > set_set_int] :
( ( R = R2 )
=> ( ( ( partia966996272515721803t_unit @ R2 )
= V )
=> ( ! [V2: set_set_int] :
( ( V = V2 )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( partia2914870419229397556t_unit @ F @ R )
= ( partia2914870419229397556t_unit @ F2 @ R2 ) ) ) ) ) ).
% partial_object.fold_congs(1)
thf(fact_10_partial__object_Ounfold__congs_I1_J,axiom,
! [R: partia4692342223508353374t_unit,R2: partia4692342223508353374t_unit,V: set_nat,F: set_nat > set_nat,F2: set_nat > set_nat] :
( ( R = R2 )
=> ( ( ( partia3499330772048238685t_unit @ R2 )
= V )
=> ( ! [V2: set_nat] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( partia6063033193446304838t_unit @ F @ R )
= ( partia6063033193446304838t_unit @ F2 @ R2 ) ) ) ) ) ).
% partial_object.unfold_congs(1)
thf(fact_11_partial__object_Ounfold__congs_I1_J,axiom,
! [R: partia4934656038542163276t_unit,R2: partia4934656038542163276t_unit,V: set_set_int,F: set_set_int > set_set_int,F2: set_set_int > set_set_int] :
( ( R = R2 )
=> ( ( ( partia966996272515721803t_unit @ R2 )
= V )
=> ( ! [V2: set_set_int] :
( ( V2 = V )
=> ( ( F @ V2 )
= ( F2 @ V2 ) ) )
=> ( ( partia2914870419229397556t_unit @ F @ R )
= ( partia2914870419229397556t_unit @ F2 @ R2 ) ) ) ) ) ).
% partial_object.unfold_congs(1)
thf(fact_12_partial__object_Oequality,axiom,
! [R: partia4692342223508353374t_unit,R2: partia4692342223508353374t_unit] :
( ( ( partia3499330772048238685t_unit @ R )
= ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( ( partia4338575005838170364t_unit @ R )
= ( partia4338575005838170364t_unit @ R2 ) )
=> ( R = R2 ) ) ) ).
% partial_object.equality
thf(fact_13_partial__object_Oequality,axiom,
! [R: partia4934656038542163276t_unit,R2: partia4934656038542163276t_unit] :
( ( ( partia966996272515721803t_unit @ R )
= ( partia966996272515721803t_unit @ R2 ) )
=> ( ( ( partia179123223730271978t_unit @ R )
= ( partia179123223730271978t_unit @ R2 ) )
=> ( R = R2 ) ) ) ).
% partial_object.equality
thf(fact_14_morphic__propE_I2_J,axiom,
! [R3: partia4692342223508353374t_unit,P: nat > $o,R: nat] :
( ( morphi2578836188448194427t_unit @ R3 @ P )
=> ( ( member_nat @ R @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( P @ R ) ) ) ).
% morphic_propE(2)
thf(fact_15_morphic__propE_I2_J,axiom,
! [R3: partia4934656038542163276t_unit,P: set_int > $o,R: set_int] :
( ( morphi7684586164509475305t_unit @ R3 @ P )
=> ( ( member_set_int @ R @ ( partia966996272515721803t_unit @ R3 ) )
=> ( P @ R ) ) ) ).
% morphic_propE(2)
thf(fact_16_zfact__iso__inv__def,axiom,
( zfact_iso_inv
= ( ^ [P2: nat] : ( hilber3159750049796175616et_int @ ( set_ord_lessThan_nat @ P2 ) @ ( ring_zfact_iso @ P2 ) ) ) ) ).
% zfact_iso_inv_def
thf(fact_17_partial__object_Oselect__convs_I1_J,axiom,
! [Carrier: set_nat,More: monoid5117177112048413924t_unit] :
( ( partia3499330772048238685t_unit @ ( partia8414553997885618600t_unit @ Carrier @ More ) )
= Carrier ) ).
% partial_object.select_convs(1)
thf(fact_18_partial__object_Oselect__convs_I1_J,axiom,
! [Carrier: set_set_int,More: monoid6410947536436496832t_unit] :
( ( partia966996272515721803t_unit @ ( partia768359127423289238t_unit @ Carrier @ More ) )
= Carrier ) ).
% partial_object.select_convs(1)
thf(fact_19_partial__object_Oselect__convs_I2_J,axiom,
! [Carrier: set_set_int,More: monoid6410947536436496832t_unit] :
( ( partia179123223730271978t_unit @ ( partia768359127423289238t_unit @ Carrier @ More ) )
= More ) ).
% partial_object.select_convs(2)
thf(fact_20_partial__object_Oselect__convs_I2_J,axiom,
! [Carrier: set_nat,More: monoid5117177112048413924t_unit] :
( ( partia4338575005838170364t_unit @ ( partia8414553997885618600t_unit @ Carrier @ More ) )
= More ) ).
% partial_object.select_convs(2)
thf(fact_21_partial__object_Oupdate__convs_I1_J,axiom,
! [Carrier2: set_set_int > set_set_int,Carrier: set_set_int,More: monoid6410947536436496832t_unit] :
( ( partia2914870419229397556t_unit @ Carrier2 @ ( partia768359127423289238t_unit @ Carrier @ More ) )
= ( partia768359127423289238t_unit @ ( Carrier2 @ Carrier ) @ More ) ) ).
% partial_object.update_convs(1)
thf(fact_22_partial__object_Oupdate__convs_I1_J,axiom,
! [Carrier2: set_nat > set_nat,Carrier: set_nat,More: monoid5117177112048413924t_unit] :
( ( partia6063033193446304838t_unit @ Carrier2 @ ( partia8414553997885618600t_unit @ Carrier @ More ) )
= ( partia8414553997885618600t_unit @ ( Carrier2 @ Carrier ) @ More ) ) ).
% partial_object.update_convs(1)
thf(fact_23_partial__object_Osurjective,axiom,
! [R: partia4692342223508353374t_unit] :
( R
= ( partia8414553997885618600t_unit @ ( partia3499330772048238685t_unit @ R ) @ ( partia4338575005838170364t_unit @ R ) ) ) ).
% partial_object.surjective
thf(fact_24_partial__object_Osurjective,axiom,
! [R: partia4934656038542163276t_unit] :
( R
= ( partia768359127423289238t_unit @ ( partia966996272515721803t_unit @ R ) @ ( partia179123223730271978t_unit @ R ) ) ) ).
% partial_object.surjective
thf(fact_25_r_Oonepideal,axiom,
princi8860937869964495385t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.onepideal
thf(fact_26_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_27_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_28_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_29_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_30_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_31_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_32_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_33_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_34_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_35_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_36_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_37_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_38_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_39_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_40_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_41_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_42_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_43_lessThan__iff,axiom,
! [I: set_int,K: set_int] :
( ( member_set_int @ I @ ( set_or5935648273017318783et_int @ K ) )
= ( ord_less_set_int @ I @ K ) ) ).
% lessThan_iff
thf(fact_44_lessThan__iff,axiom,
! [I: nat > set_int,K: nat > set_int] :
( ( member_nat_set_int @ I @ ( set_or7260056672446632558et_int @ K ) )
= ( ord_less_nat_set_int @ I @ K ) ) ).
% lessThan_iff
thf(fact_45_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_46_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_47_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_48_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_49_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_50_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_51_ring__iso__memE_I1_J,axiom,
! [H: nat > nat,R3: partia4692342223508353374t_unit,S2: partia4692342223508353374t_unit,X: nat] :
( ( member_nat_nat @ H @ ( ring_i3628375019190340560t_unit @ R3 @ S2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_nat @ ( H @ X ) @ ( partia3499330772048238685t_unit @ S2 ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_52_ring__iso__memE_I1_J,axiom,
! [H: nat > set_int,R3: partia4692342223508353374t_unit,S2: partia4934656038542163276t_unit,X: nat] :
( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R3 @ S2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( member_set_int @ ( H @ X ) @ ( partia966996272515721803t_unit @ S2 ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_53_ring__iso__memE_I1_J,axiom,
! [H: set_int > nat,R3: partia4934656038542163276t_unit,S2: partia4692342223508353374t_unit,X: set_int] :
( ( member_set_int_nat @ H @ ( ring_i6162119212153773794t_unit @ R3 @ S2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_nat @ ( H @ X ) @ ( partia3499330772048238685t_unit @ S2 ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_54_ring__iso__memE_I1_J,axiom,
! [H: set_int > set_int,R3: partia4934656038542163276t_unit,S2: partia4934656038542163276t_unit,X: set_int] :
( ( member5205197933313416826et_int @ H @ ( ring_i7857697894474155344t_unit @ R3 @ S2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( member_set_int @ ( H @ X ) @ ( partia966996272515721803t_unit @ S2 ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_55_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_56_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_57_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_58_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_59_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_60_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A2: set_set_int] :
( ( collect_set_int
@ ^ [X2: set_int] : ( member_set_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A2: set_nat_set_int] :
( ( collect_nat_set_int
@ ^ [X2: nat > set_int] : ( member_nat_set_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_63_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_64_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_65_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_66_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_67_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_68_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_69_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_70_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_71_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_72_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_73_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_74_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_75_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_76_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_77_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_78_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_79_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_80_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_81_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_82_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_83_r_Ocgenideal__is__principalideal,axiom,
! [I: set_int] :
( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( princi8860937869964495385t_unit @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.cgenideal_is_principalideal
thf(fact_84_r_Osemiring__axioms,axiom,
semiri8708897239777792527t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.semiring_axioms
thf(fact_85_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_86_r_Ocarrier__is__subcring,axiom,
subcri1024317279029940167t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.carrier_is_subcring
thf(fact_87_r_Ocgenideal__self,axiom,
! [I: set_int] :
( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ I @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I ) ) ) ).
% r.cgenideal_self
thf(fact_88_r_Ocarrier__not__empty,axiom,
( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= bot_bot_set_set_int ) ).
% r.carrier_not_empty
thf(fact_89_r_Oabelian__monoid__axioms,axiom,
abelia3815030880812984441t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.abelian_monoid_axioms
thf(fact_90_r_Oadd_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 ) ) ) ) ) ).
% r.add.r_cancel
thf(fact_91_r_Oadd_Om__lcomm,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( 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 ) ) @ X @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Z ) ) ) ) ) ) ).
% r.add.m_lcomm
thf(fact_92_r_Oadd_Om__comm,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X ) ) ) ) ).
% r.add.m_comm
thf(fact_93_r_Oadd_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 ) ) ) ) ) ).
% r.add.l_cancel
thf(fact_94_r_Oadd_Om__assoc,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( 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 ) ) @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) ) ) ) ) ) ).
% r.add.m_assoc
thf(fact_95_r_Oadd_Om__closed,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( 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 ) ) @ X @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.m_closed
thf(fact_96_r_Oadd_Oright__cancel,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( 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 @ X )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% r.add.right_cancel
thf(fact_97_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_98_ring__iso__memE_I3_J,axiom,
! [H: nat > set_int,R3: partia4692342223508353374t_unit,S2: partia4934656038542163276t_unit,X: nat,Y: nat] :
( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R3 @ S2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( H @ ( add_nat_Product_unit @ R3 @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_99_ring__iso__memE_I3_J,axiom,
! [H: set_int > set_int,R3: partia4934656038542163276t_unit,S2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( member5205197933313416826et_int @ H @ ( ring_i7857697894474155344t_unit @ R3 @ S2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( H @ ( add_se5859248395121729892t_unit @ R3 @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_100_morphic__propE_I4_J,axiom,
! [R3: partia4692342223508353374t_unit,P: nat > $o,R1: nat,R22: nat] :
( ( morphi2578836188448194427t_unit @ R3 @ P )
=> ( ( member_nat @ R1 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ R22 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( P @ ( add_nat_Product_unit @ R3 @ R1 @ R22 ) ) ) ) ) ).
% morphic_propE(4)
thf(fact_101_morphic__propE_I4_J,axiom,
! [R3: partia4934656038542163276t_unit,P: set_int > $o,R1: set_int,R22: set_int] :
( ( morphi7684586164509475305t_unit @ R3 @ P )
=> ( ( member_set_int @ R1 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ R22 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( P @ ( add_se5859248395121729892t_unit @ R3 @ R1 @ R22 ) ) ) ) ) ).
% morphic_propE(4)
thf(fact_102_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_103_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_104_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_105_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_106_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_107_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_108_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_109_r_Oadd_Oint__pow__distrib,axiom,
! [X: set_int,Y: set_int,I: int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ Y ) ) ) ) ) ).
% r.add.int_pow_distrib
thf(fact_110_r_Oadd_Oint__pow__mult__distrib,axiom,
! [X: set_int,Y: set_int,I: int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ Y ) ) ) ) ) ) ).
% r.add.int_pow_mult_distrib
thf(fact_111_r_Oadd_Oinv__comm,axiom,
! [X: set_int,Y: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.add.inv_comm
thf(fact_112_r_Oadd_Ol__inv__ex,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [X3: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X3 @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.l_inv_ex
thf(fact_113_r_Oadd_Oone__unique,axiom,
! [U: set_int] :
( ( member_set_int @ U @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ! [X3: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ U @ X3 )
= X3 ) )
=> ( U
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.one_unique
thf(fact_114_r_Oadd_Or__inv__ex,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [X3: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ X3 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.r_inv_ex
thf(fact_115_r_Ominus__unique,axiom,
! [Y: set_int,X: set_int,Y2: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( 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 ) ) ) ) ) ) ).
% r.minus_unique
thf(fact_116_r_Ol__distr,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( 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 ) ) @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Z ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) ) ) ) ) ) ).
% r.l_distr
thf(fact_117_r_Or__distr,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( 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 ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ X ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ Y ) ) ) ) ) ) ).
% r.r_distr
thf(fact_118_r_Oadd_Ogroup__commutes__pow,axiom,
! [X: set_int,Y: set_int,N: nat] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ X ) @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ X ) ) ) ) ) ) ).
% r.add.group_commutes_pow
thf(fact_119_r_Om__lcomm,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( 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 ) ) @ X @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Z ) ) ) ) ) ) ).
% r.m_lcomm
thf(fact_120_r_Om__comm,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X ) ) ) ) ).
% r.m_comm
thf(fact_121_r_Om__assoc,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( 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 ) ) @ X @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) ) ) ) ) ) ).
% r.m_assoc
thf(fact_122_r_Oadd_Opow__mult__distrib,axiom,
! [X: set_int,Y: set_int,N: nat] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ X ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ Y ) ) ) ) ) ) ).
% r.add.pow_mult_distrib
thf(fact_123_r_Oadd_Onat__pow__distrib,axiom,
! [X: set_int,Y: set_int,N: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ X ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ Y ) ) ) ) ) ).
% r.add.nat_pow_distrib
thf(fact_124_r_Oadd_Onat__pow__comm,axiom,
! [X: set_int,N: nat,M: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ X ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ X ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ X ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ X ) ) ) ) ).
% r.add.nat_pow_comm
thf(fact_125_r_Oadd__pow__rdistr,axiom,
! [A: set_int,B: set_int,K: nat] :
( ( 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_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ B ) )
= ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ) ) ) ).
% r.add_pow_rdistr
thf(fact_126_r_Oadd__pow__ldistr,axiom,
! [A: set_int,B: set_int,K: nat] :
( ( 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_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ A ) @ B )
= ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ) ) ) ).
% r.add_pow_ldistr
thf(fact_127_r_Oadd__pow__rdistr__int,axiom,
! [A: set_int,B: set_int,K: 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 ) ) @ K @ B ) )
= ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ) ) ) ).
% r.add_pow_rdistr_int
thf(fact_128_r_Oadd__pow__ldistr__int,axiom,
! [A: set_int,B: set_int,K: 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 ) ) @ K @ A ) @ B )
= ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ) ) ) ).
% r.add_pow_ldistr_int
thf(fact_129_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_130_r_Ozero__closed,axiom,
member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% r.zero_closed
thf(fact_131_r_Om__closed,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( 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 ) ) @ X @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.m_closed
thf(fact_132_r_Oadd_Onat__pow__closed,axiom,
! [X: set_int,N: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ X ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.add.nat_pow_closed
thf(fact_133_r_Oadd_Onat__pow__one,axiom,
! [N: nat] :
( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.add.nat_pow_one
thf(fact_134_r_Oadd_Oint__pow__1,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ one_one_int @ X )
= X ) ) ).
% r.add.int_pow_1
thf(fact_135_r_Oadd_Oint__pow__closed,axiom,
! [X: set_int,I: int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.add.int_pow_closed
thf(fact_136_r_Oadd_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 ) ) ) ) ).
% r.add.int_pow_one
thf(fact_137_r_Or__zero,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= X ) ) ).
% r.r_zero
thf(fact_138_r_Ol__zero,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X )
= X ) ) ).
% r.l_zero
thf(fact_139_r_Oadd_Or__cancel__one_H,axiom,
! [X: set_int,A: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( X
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ X ) )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.add.r_cancel_one'
thf(fact_140_r_Oadd_Or__cancel__one,axiom,
! [X: set_int,A: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ X )
= X )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.add.r_cancel_one
thf(fact_141_r_Oadd_Ol__cancel__one_H,axiom,
! [X: set_int,A: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( X
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ A ) )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.add.l_cancel_one'
thf(fact_142_r_Oadd_Ol__cancel__one,axiom,
! [X: set_int,A: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ A )
= X )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.add.l_cancel_one
thf(fact_143_r_Or__null,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.r_null
thf(fact_144_r_Ol__null,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.l_null
thf(fact_145_r_Oadd_Onat__pow__0,axiom,
! [X: set_int] :
( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ zero_zero_nat @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.add.nat_pow_0
thf(fact_146_r_Oadd_Onat__pow__eone,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ one_one_nat @ X )
= X ) ) ).
% r.add.nat_pow_eone
thf(fact_147_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_148_ring__iso__memE_I2_J,axiom,
! [H: nat > set_int,R3: partia4692342223508353374t_unit,S2: partia4934656038542163276t_unit,X: nat,Y: nat] :
( ( member_nat_set_int @ H @ ( ring_i1863809825068120638t_unit @ R3 @ S2 ) )
=> ( ( member_nat @ X @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ Y @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( H @ ( mult_n6028127365542633569t_unit @ R3 @ X @ Y ) )
= ( mult_s3864001451298473021t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_149_ring__iso__memE_I2_J,axiom,
! [H: set_int > set_int,R3: partia4934656038542163276t_unit,S2: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
( ( member5205197933313416826et_int @ H @ ( ring_i7857697894474155344t_unit @ R3 @ S2 ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( H @ ( mult_s3864001451298473021t_unit @ R3 @ X @ Y ) )
= ( mult_s3864001451298473021t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_150_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_151_morphic__propE_I3_J,axiom,
! [R3: partia4692342223508353374t_unit,P: nat > $o,R1: nat,R22: nat] :
( ( morphi2578836188448194427t_unit @ R3 @ P )
=> ( ( member_nat @ R1 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ R22 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( P @ ( mult_n6028127365542633569t_unit @ R3 @ R1 @ R22 ) ) ) ) ) ).
% morphic_propE(3)
thf(fact_152_morphic__propE_I3_J,axiom,
! [R3: partia4934656038542163276t_unit,P: set_int > $o,R1: set_int,R22: set_int] :
( ( morphi7684586164509475305t_unit @ R3 @ P )
=> ( ( member_set_int @ R1 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ R22 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( P @ ( mult_s3864001451298473021t_unit @ R3 @ R1 @ R22 ) ) ) ) ) ).
% morphic_propE(3)
thf(fact_153_zfact__iso__0,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ring_zfact_iso @ N @ zero_zero_nat )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% zfact_iso_0
thf(fact_154_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_155_r_OboundD__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 ) ) ) ) ) ) ).
% r.boundD_carrier
thf(fact_156_r_Omonoid__cancelI,axiom,
( ! [A3: set_int,B2: set_int,C2: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C2 @ A3 )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C2 @ B2 ) )
=> ( ( member_set_int @ A3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: set_int,B2: set_int,C2: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A3 @ C2 )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B2 @ C2 ) )
=> ( ( member_set_int @ A3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid497721730651901107t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.monoid_cancelI
thf(fact_157_r_Ozeropideal,axiom,
princi8860937869964495385t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.zeropideal
thf(fact_158_r_Oadd_Oint__pow__mult,axiom,
! [X: set_int,I: int,J: int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( plus_plus_int @ I @ J ) @ X )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ J @ X ) ) ) ) ).
% r.add.int_pow_mult
thf(fact_159_r_Oline__extension__mem__iff,axiom,
! [U: set_int,K2: set_set_int,A: set_int,E: set_set_int] :
( ( member_set_int @ U @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ A @ E ) )
= ( ? [X2: set_int] :
( ( member_set_int @ X2 @ K2 )
& ? [Y3: set_int] :
( ( member_set_int @ Y3 @ E )
& ( U
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ A ) @ Y3 ) ) ) ) ) ) ).
% r.line_extension_mem_iff
thf(fact_160_r_Oadd_Onat__pow__mult,axiom,
! [X: set_int,N: nat,M: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ X ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ X ) )
= ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( plus_plus_nat @ N @ M ) @ X ) ) ) ).
% r.add.nat_pow_mult
thf(fact_161_r_Oadd_Opow__eq__div2,axiom,
! [X: set_int,M: nat,N: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ X )
= ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ X ) )
=> ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( minus_minus_nat @ M @ N ) @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.pow_eq_div2
thf(fact_162_r_Oadd_Onat__pow__Suc2,axiom,
! [X: set_int,N: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( suc @ N ) @ X )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ X ) ) ) ) ).
% r.add.nat_pow_Suc2
thf(fact_163_r_Ocgenideal__prod,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 ) ) ) )
=> ( ( set_mu2785919024023201382t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B ) )
= ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) ) ) ) ) ).
% r.cgenideal_prod
thf(fact_164_r_Oadd_Oint__pow__pow,axiom,
! [X: set_int,M: int,N: int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ X ) )
= ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( times_times_int @ N @ M ) @ X ) ) ) ).
% r.add.int_pow_pow
thf(fact_165_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_166_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_167_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_168_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_169_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_170_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_171_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_172_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_173_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_174_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_175_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_176_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_177_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_178_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_179_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_180_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_181_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_182_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_183_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_184_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_185_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_186_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_187_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_188_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_189_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_190_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_191_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_192_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_193_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_194_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_195_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_196_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_197_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_198_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_199_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_200_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_201_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_202_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_203_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_204_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_205_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_206_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_207_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_208_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_209_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_210_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_211_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_212_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_213_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_214_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_215_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_216_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_217_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_218_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_219_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_220_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_221_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_222_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_223_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_224_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_225_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_226_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_227_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_228_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_229_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_230_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_231_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_232_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_233_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_234_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_235_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_236_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_237_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_238_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_239_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_240_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_241_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_242_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_243_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_244_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_245_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_246_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_247_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_248_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_249_r_Oadd_Onat__pow__Suc,axiom,
! [N: nat,X: set_int] :
( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( suc @ N ) @ X )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ X ) @ X ) ) ).
% r.add.nat_pow_Suc
thf(fact_250_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_251_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_252_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_253_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_254_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_255_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q ) ) ) ) ).
% less_natE
thf(fact_256_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_257_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_258_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M4 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_259_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K4: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K4 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_260_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_261_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_262_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_263_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_264_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_265_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_266_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_267_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_268_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_269_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_270_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_271_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_272_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_273_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_274_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_275_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_276_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_277_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_278_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_279_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_280_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_281_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_282_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_283_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_284_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_285_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_286_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_287_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_288_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_289_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_290_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_291_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_292_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B3: int] : ( times_times_int @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_293_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B3: nat] : ( times_times_nat @ B3 @ A4 ) ) ) ).
% mult.commute
thf(fact_294_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_295_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_296_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_297_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_298_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_299_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_300_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_301_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_302_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_303_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_304_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_305_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_306_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_307_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_308_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_309_group__cancel_Oadd2,axiom,
! [B4: int,K: int,B: int,A: int] :
( ( B4
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B4 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_310_group__cancel_Oadd2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_311_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_312_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_313_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_314_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_315_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_316_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_317_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_318_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_319_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_320_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_321_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_322_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_323_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_324_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_325_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A4: int,B3: int] :
( ( minus_minus_int @ A4 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_326_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_327_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_328_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_329_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_330_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_331_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_332_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_333_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_334_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_335_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_336_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_337_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_338_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_339_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_340_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_341_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_342_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_343_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_344_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_345_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_346_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_347_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_348_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_349_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_350_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_351_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_352_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_353_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_354_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_355_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_356_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_357_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_358_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_359_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_360_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_361_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_362_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_363_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y5: nat] : ( P @ zero_zero_nat @ ( suc @ Y5 ) )
=> ( ! [X3: nat,Y5: nat] :
( ( P @ X3 @ Y5 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y5 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_364_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_365_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_366_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_367_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_368_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_369_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_370_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_371_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_372_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_373_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_374_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_375_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_376_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_377_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_378_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_379_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_380_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_381_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_382_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_383_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_384_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_385_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K4: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K4 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K4 )
=> ( P @ I3 @ K4 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_386_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_387_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_388_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_389_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_390_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_391_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_392_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_393_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_394_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_395_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_396_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_397_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_398_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_399_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_400_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_401_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_402_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_403_bound__def,axiom,
( bound_set_int
= ( ^ [Z3: set_int,N3: nat,F3: nat > set_int] :
! [M4: nat] :
( ( ord_less_nat @ N3 @ M4 )
=> ( ( F3 @ M4 )
= Z3 ) ) ) ) ).
% bound_def
thf(fact_404_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_405_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_406_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_407_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_408_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_409_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_410_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_411_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_412_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_413_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_414_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_415_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_416_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_417_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_418_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_419_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_420_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_421_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_422_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_423_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_424_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_425_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_426_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_427_r_Ocgenideal__eq__genideal,axiom,
! [I: set_int] :
( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I )
= ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ I @ bot_bot_set_set_int ) ) ) ) ).
% r.cgenideal_eq_genideal
thf(fact_428_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_429_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_430_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_431_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_432_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_433_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_434_r_Ogenideal__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 ) ) ).
% r.genideal_zero
thf(fact_435_r_Ogenideal__self_H,axiom,
! [I: set_int] :
( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ I @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ I @ bot_bot_set_set_int ) ) ) ) ).
% r.genideal_self'
thf(fact_436_r_Oadd_Onat__pow__pow,axiom,
! [X: set_int,M: nat,N: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ X ) )
= ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( times_times_nat @ N @ M ) @ X ) ) ) ).
% r.add.nat_pow_pow
thf(fact_437_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_438_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_439_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_440_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_441_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_442_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_443_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_444_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_445_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_446_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_447_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_448_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_449_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_450_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_451_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_452_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_453_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_454_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_455_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_456_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_457_single__Diff__lessThan,axiom,
! [K: set_int] :
( ( minus_8897228262479074673et_int @ ( insert_set_int @ K @ bot_bot_set_set_int ) @ ( set_or5935648273017318783et_int @ K ) )
= ( insert_set_int @ K @ bot_bot_set_set_int ) ) ).
% single_Diff_lessThan
thf(fact_458_single__Diff__lessThan,axiom,
! [K: nat] :
( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
= ( insert_nat @ K @ bot_bot_set_nat ) ) ).
% single_Diff_lessThan
thf(fact_459_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_460_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_461_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_462_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_463_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_464_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_465_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_466_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_467_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_468_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_469_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_470_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_471_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_472_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_473_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_474_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_475_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_476_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_477_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_478_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_479_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_480_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_481_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_482_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_483_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_484_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_485_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_486_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_487_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_488_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_489_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_490_combine__common__factor,axiom,
! [A: int,E2: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_491_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_492_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_493_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_494_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_495_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_496_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_497_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_498_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_499_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_500_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_501_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_502_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_503_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_504_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_505_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_506_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_507_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_508_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_509_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_510_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_511_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_512_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_513_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_514_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_515_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_516_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_517_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_518_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_519_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_520_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_521_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_522_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_523_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_524_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_525_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_526_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_527_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_528_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_529_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_530_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_531_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_532_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_533_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_534_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_535_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_536_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_537_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_538_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_539_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_540_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_541_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_542_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_543_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_544_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_545_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_546_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_547_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_548_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_549_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_550_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_551_eq__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_552_eq__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_553_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_554_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_555_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_556_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_557_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_558_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_559_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_560_less__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_561_less__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% less_add_iff2
thf(fact_562_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_563_r_OIdl__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 ) ) ) ) ) ) ).
% r.Idl_subset_ideal'
thf(fact_564_r_Obound__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 ) ) ).
% r.bound_upD
thf(fact_565_r_Odimension_Osimps,axiom,
! [A1: nat,A22: set_set_int,A32: set_set_int] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A1 @ A22 @ A32 )
= ( ? [K5: set_set_int] :
( ( A1 = zero_zero_nat )
& ( A22 = K5 )
& ( A32
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
| ? [V3: set_int,E3: set_set_int,N3: nat,K5: set_set_int] :
( ( A1
= ( suc @ N3 ) )
& ( A22 = K5 )
& ( A32
= ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K5 @ V3 @ E3 ) )
& ( member_set_int @ V3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ~ ( member_set_int @ V3 @ E3 )
& ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N3 @ K5 @ E3 ) ) ) ) ).
% r.dimension.simps
thf(fact_566_r_Oline__extension__in__carrier,axiom,
! [K2: set_set_int,A: set_int,E: set_set_int] :
( ( ord_le4403425263959731960et_int @ K2 @ ( 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 ) ) @ K2 @ A @ E ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.line_extension_in_carrier
thf(fact_567_r_Ogenideal__self,axiom,
! [S2: set_set_int] :
( ( ord_le4403425263959731960et_int @ S2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ S2 @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S2 ) ) ) ).
% r.genideal_self
thf(fact_568_r_Osubset__Idl__subset,axiom,
! [I4: set_set_int,H2: set_set_int] :
( ( ord_le4403425263959731960et_int @ I4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ H2 @ I4 )
=> ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 ) @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I4 ) ) ) ) ).
% r.subset_Idl_subset
thf(fact_569_r_Oset__mult__closed,axiom,
! [H2: set_set_int,K2: set_set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( set_mu2785919024023201382t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 @ K2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.set_mult_closed
thf(fact_570_r_OSuc__dim,axiom,
! [V4: set_int,E: set_set_int,N: nat,K2: 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 @ K2 @ E )
=> ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( suc @ N ) @ K2 @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ V4 @ E ) ) ) ) ) ).
% r.Suc_dim
thf(fact_571_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_572_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_573_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_574_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_575_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_576_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_577_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_578_r_Ozero__dim,axiom,
! [K2: set_set_int] : ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ zero_zero_nat @ K2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) ).
% r.zero_dim
thf(fact_579_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_580_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_581_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_582_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_583_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_584_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_585_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_586_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_587_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_588_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_589_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_590_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_591_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_592_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_593_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_594_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_595_r_Odimension_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,E4: set_set_int,N2: nat] :
( ( A1
= ( suc @ N2 ) )
=> ( ( A32
= ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A22 @ V2 @ E4 ) )
=> ( ( member_set_int @ V2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ~ ( member_set_int @ V2 @ E4 )
=> ~ ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ A22 @ E4 ) ) ) ) ) ) ) ).
% r.dimension.cases
thf(fact_596_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_597_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_598_mem__upI,axiom,
! [F: nat > nat,R3: partia4692342223508353374t_unit] :
( ! [N2: nat] : ( member_nat @ ( F @ N2 ) @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ? [N5: nat] : ( bound_nat @ ( zero_n5149899317435570679t_unit @ R3 ) @ N5 @ F )
=> ( member_nat_nat @ F @ ( up_nat_Product_unit @ R3 ) ) ) ) ).
% mem_upI
thf(fact_599_mem__upI,axiom,
! [F: nat > set_int,R3: partia4934656038542163276t_unit] :
( ! [N2: nat] : ( member_set_int @ ( F @ N2 ) @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ? [N5: nat] : ( bound_set_int @ ( zero_s6269048424454532197t_unit @ R3 ) @ N5 @ F )
=> ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ R3 ) ) ) ) ).
% mem_upI
thf(fact_600_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_601_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_602_lift__Suc__mono__le,axiom,
! [F: nat > set_set_int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_le4403425263959731960et_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le4403425263959731960et_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_603_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_604_lift__Suc__antimono__le,axiom,
! [F: nat > set_set_int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_le4403425263959731960et_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le4403425263959731960et_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_605_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_606_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_607_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_608_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_609_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_610_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_611_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_612_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_613_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_614_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_615_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_616_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_617_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_618_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_619_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_620_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_621_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_622_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_623_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_624_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_625_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_626_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_627_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_628_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_629_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_630_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_631_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_632_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_633_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_634_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_635_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_636_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_637_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_638_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_639_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_640_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_641_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_642_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_643_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_644_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_645_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_646_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_647_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_648_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_649_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_650_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_651_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_652_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_653_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_654_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_655_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_656_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_657_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_658_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_659_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_660_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_661_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_662_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_663_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_664_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_665_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_666_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_667_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_668_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_669_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_670_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_671_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_672_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_673_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_674_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_675_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_676_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_677_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_678_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_679_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_680_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_681_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_682_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_683_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_684_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_685_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_686_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_687_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_688_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_689_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_690_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_691_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_692_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_693_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_694_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_695_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_696_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_697_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_698_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_699_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_700_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_701_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_702_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_703_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_704_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_705_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_706_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_707_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_708_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_709_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_710_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_711_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_712_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_713_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_714_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_715_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_716_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_717_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_718_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_719_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_720_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_721_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_722_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_723_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_724_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_725_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_726_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_727_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_728_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_729_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_730_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_731_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_732_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_733_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_734_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_735_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_736_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_737_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_738_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_739_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_740_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_741_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_742_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_743_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_744_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_745_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_746_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_747_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_748_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_749_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_750_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_751_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_752_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_753_ordered__ring__class_Ole__add__iff1,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_754_ordered__ring__class_Ole__add__iff2,axiom,
! [A: int,E2: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_755_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V4: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V4 )
=> ( ( ( plus_plus_int @ U @ V4 )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V4 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_756_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_757_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_758_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_759_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_760_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_761_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_762_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_763_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_764_convex__bound__lt,axiom,
! [X: int,A: int,Y: int,U: int,V4: int] :
( ( ord_less_int @ X @ A )
=> ( ( ord_less_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V4 )
=> ( ( ( plus_plus_int @ U @ V4 )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V4 @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_765_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_766_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_767_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_768_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_769_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_770_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_771_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B4: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B4 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_772_Diff__eq__empty__iff,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ( minus_8897228262479074673et_int @ A2 @ B4 )
= bot_bot_set_set_int )
= ( ord_le4403425263959731960et_int @ A2 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_773_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_774_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_775_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_776_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_777_empty__Collect__eq,axiom,
! [P: set_int > $o] :
( ( bot_bot_set_set_int
= ( collect_set_int @ P ) )
= ( ! [X2: set_int] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_778_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_779_Collect__empty__eq,axiom,
! [P: set_int > $o] :
( ( ( collect_set_int @ P )
= bot_bot_set_set_int )
= ( ! [X2: set_int] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_780_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_781_all__not__in__conv,axiom,
! [A2: set_nat_set_int] :
( ( ! [X2: nat > set_int] :
~ ( member_nat_set_int @ X2 @ A2 ) )
= ( A2 = bot_bo8417611410066262939et_int ) ) ).
% all_not_in_conv
thf(fact_782_all__not__in__conv,axiom,
! [A2: set_set_int] :
( ( ! [X2: set_int] :
~ ( member_set_int @ X2 @ A2 ) )
= ( A2 = bot_bot_set_set_int ) ) ).
% all_not_in_conv
thf(fact_783_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_784_empty__iff,axiom,
! [C: nat > set_int] :
~ ( member_nat_set_int @ C @ bot_bo8417611410066262939et_int ) ).
% empty_iff
thf(fact_785_empty__iff,axiom,
! [C: set_int] :
~ ( member_set_int @ C @ bot_bot_set_set_int ) ).
% empty_iff
thf(fact_786_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_787_insert__absorb2,axiom,
! [X: set_int,A2: set_set_int] :
( ( insert_set_int @ X @ ( insert_set_int @ X @ A2 ) )
= ( insert_set_int @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_788_insert__absorb2,axiom,
! [X: nat,A2: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ X @ A2 ) )
= ( insert_nat @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_789_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_790_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_791_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_792_insertCI,axiom,
! [A: nat,B4: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B4 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).
% insertCI
thf(fact_793_insertCI,axiom,
! [A: set_int,B4: set_set_int,B: set_int] :
( ( ~ ( member_set_int @ A @ B4 )
=> ( A = B ) )
=> ( member_set_int @ A @ ( insert_set_int @ B @ B4 ) ) ) ).
% insertCI
thf(fact_794_insertCI,axiom,
! [A: nat > set_int,B4: set_nat_set_int,B: nat > set_int] :
( ( ~ ( member_nat_set_int @ A @ B4 )
=> ( A = B ) )
=> ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ B4 ) ) ) ).
% insertCI
thf(fact_795_Diff__idemp,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( minus_8897228262479074673et_int @ ( minus_8897228262479074673et_int @ A2 @ B4 ) @ B4 )
= ( minus_8897228262479074673et_int @ A2 @ B4 ) ) ).
% Diff_idemp
thf(fact_796_Diff__iff,axiom,
! [C: nat,A2: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B4 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_797_Diff__iff,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B4 ) )
= ( ( member_nat_set_int @ C @ A2 )
& ~ ( member_nat_set_int @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_798_Diff__iff,axiom,
! [C: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A2 @ B4 ) )
= ( ( member_set_int @ C @ A2 )
& ~ ( member_set_int @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_799_DiffI,axiom,
! [C: nat,A2: set_nat,B4: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B4 ) ) ) ) ).
% DiffI
thf(fact_800_DiffI,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ C @ A2 )
=> ( ~ ( member_nat_set_int @ C @ B4 )
=> ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B4 ) ) ) ) ).
% DiffI
thf(fact_801_DiffI,axiom,
! [C: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ A2 )
=> ( ~ ( member_set_int @ C @ B4 )
=> ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A2 @ B4 ) ) ) ) ).
% DiffI
thf(fact_802_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_803_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_804_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_805_empty__subsetI,axiom,
! [A2: set_set_int] : ( ord_le4403425263959731960et_int @ bot_bot_set_set_int @ A2 ) ).
% empty_subsetI
thf(fact_806_insert__subset,axiom,
! [X: nat,A2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B4 )
= ( ( member_nat @ X @ B4 )
& ( ord_less_eq_set_nat @ A2 @ B4 ) ) ) ).
% insert_subset
thf(fact_807_insert__subset,axiom,
! [X: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ ( insert_nat_set_int @ X @ A2 ) @ B4 )
= ( ( member_nat_set_int @ X @ B4 )
& ( ord_le5995675665013768039et_int @ A2 @ B4 ) ) ) ).
% insert_subset
thf(fact_808_insert__subset,axiom,
! [X: set_int,A2: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( insert_set_int @ X @ A2 ) @ B4 )
= ( ( member_set_int @ X @ B4 )
& ( ord_le4403425263959731960et_int @ A2 @ B4 ) ) ) ).
% insert_subset
thf(fact_809_singletonI,axiom,
! [A: nat > set_int] : ( member_nat_set_int @ A @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ) ).
% singletonI
thf(fact_810_singletonI,axiom,
! [A: set_int] : ( member_set_int @ A @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ).
% singletonI
thf(fact_811_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_812_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_813_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_814_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_815_Diff__cancel,axiom,
! [A2: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ A2 )
= bot_bot_set_set_int ) ).
% Diff_cancel
thf(fact_816_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_817_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_818_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_819_Diff__empty,axiom,
! [A2: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ bot_bot_set_set_int )
= A2 ) ).
% Diff_empty
thf(fact_820_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_821_insert__Diff1,axiom,
! [X: nat,B4: set_nat,A2: set_nat] :
( ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B4 )
= ( minus_minus_set_nat @ A2 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_822_insert__Diff1,axiom,
! [X: nat > set_int,B4: set_nat_set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ X @ B4 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X @ A2 ) @ B4 )
= ( minus_3247115583872269408et_int @ A2 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_823_insert__Diff1,axiom,
! [X: set_int,B4: set_set_int,A2: set_set_int] :
( ( member_set_int @ X @ B4 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X @ A2 ) @ B4 )
= ( minus_8897228262479074673et_int @ A2 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_824_Diff__insert0,axiom,
! [X: nat,A2: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B4 ) )
= ( minus_minus_set_nat @ A2 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_825_Diff__insert0,axiom,
! [X: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ~ ( member_nat_set_int @ X @ A2 )
=> ( ( minus_3247115583872269408et_int @ A2 @ ( insert_nat_set_int @ X @ B4 ) )
= ( minus_3247115583872269408et_int @ A2 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_826_Diff__insert0,axiom,
! [X: set_int,A2: set_set_int,B4: set_set_int] :
( ~ ( member_set_int @ X @ A2 )
=> ( ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ X @ B4 ) )
= ( minus_8897228262479074673et_int @ A2 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_827_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_828_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_829_psubsetI,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ( A2 != B4 )
=> ( ord_less_set_set_int @ A2 @ B4 ) ) ) ).
% psubsetI
thf(fact_830_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_831_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_832_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_833_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_834_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_835_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_836_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_837_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_838_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_839_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_840_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M6: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M6 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_841_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_842_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_843_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_844_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_845_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_846_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_847_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_848_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_849_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_850_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_851_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y5: nat,Z4: nat] :
( ( R3 @ X3 @ Y5 )
=> ( ( R3 @ Y5 @ Z4 )
=> ( R3 @ X3 @ Z4 ) ) )
=> ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
=> ( R3 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_852_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_853_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_854_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_855_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_856_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_857_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_858_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_859_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_860_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_861_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_862_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_863_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_864_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_865_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_866_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_867_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_868_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_869_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_870_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_871_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_872_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_873_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_874_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_875_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_876_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_877_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_878_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_879_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_880_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_881_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_882_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_883_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_884_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_885_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_886_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_887_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_888_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_889_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_890_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_891_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_892_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_893_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_894_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_895_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_896_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_897_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_898_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_899_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_900_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_901_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_902_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_903_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_904_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_905_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_906_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_907_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_908_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_909_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_910_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_911_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_912_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_913_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K4 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_914_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_915_bot__set__def,axiom,
( bot_bot_set_set_int
= ( collect_set_int @ bot_bot_set_int_o ) ) ).
% bot_set_def
thf(fact_916_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_917_ex__in__conv,axiom,
! [A2: set_nat_set_int] :
( ( ? [X2: nat > set_int] : ( member_nat_set_int @ X2 @ A2 ) )
= ( A2 != bot_bo8417611410066262939et_int ) ) ).
% ex_in_conv
thf(fact_918_ex__in__conv,axiom,
! [A2: set_set_int] :
( ( ? [X2: set_int] : ( member_set_int @ X2 @ A2 ) )
= ( A2 != bot_bot_set_set_int ) ) ).
% ex_in_conv
thf(fact_919_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_920_equals0I,axiom,
! [A2: set_nat_set_int] :
( ! [Y5: nat > set_int] :
~ ( member_nat_set_int @ Y5 @ A2 )
=> ( A2 = bot_bo8417611410066262939et_int ) ) ).
% equals0I
thf(fact_921_equals0I,axiom,
! [A2: set_set_int] :
( ! [Y5: set_int] :
~ ( member_set_int @ Y5 @ A2 )
=> ( A2 = bot_bot_set_set_int ) ) ).
% equals0I
thf(fact_922_equals0I,axiom,
! [A2: set_nat] :
( ! [Y5: nat] :
~ ( member_nat @ Y5 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_923_equals0D,axiom,
! [A2: set_nat_set_int,A: nat > set_int] :
( ( A2 = bot_bo8417611410066262939et_int )
=> ~ ( member_nat_set_int @ A @ A2 ) ) ).
% equals0D
thf(fact_924_equals0D,axiom,
! [A2: set_set_int,A: set_int] :
( ( A2 = bot_bot_set_set_int )
=> ~ ( member_set_int @ A @ A2 ) ) ).
% equals0D
thf(fact_925_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_926_emptyE,axiom,
! [A: nat > set_int] :
~ ( member_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ).
% emptyE
thf(fact_927_emptyE,axiom,
! [A: set_int] :
~ ( member_set_int @ A @ bot_bot_set_set_int ) ).
% emptyE
thf(fact_928_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_929_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_930_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_931_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_932_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_933_insert__commute,axiom,
! [X: set_int,Y: set_int,A2: set_set_int] :
( ( insert_set_int @ X @ ( insert_set_int @ Y @ A2 ) )
= ( insert_set_int @ Y @ ( insert_set_int @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_934_insert__commute,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ Y @ A2 ) )
= ( insert_nat @ Y @ ( insert_nat @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_935_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B4: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B4 )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ B4 ) )
= ( ( ( A = B )
=> ( A2 = B4 ) )
& ( ( A != B )
=> ? [C4: set_nat] :
( ( A2
= ( insert_nat @ B @ C4 ) )
& ~ ( member_nat @ B @ C4 )
& ( B4
= ( insert_nat @ A @ C4 ) )
& ~ ( member_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_936_insert__eq__iff,axiom,
! [A: set_int,A2: set_set_int,B: set_int,B4: set_set_int] :
( ~ ( member_set_int @ A @ A2 )
=> ( ~ ( member_set_int @ B @ B4 )
=> ( ( ( insert_set_int @ A @ A2 )
= ( insert_set_int @ B @ B4 ) )
= ( ( ( A = B )
=> ( A2 = B4 ) )
& ( ( A != B )
=> ? [C4: set_set_int] :
( ( A2
= ( insert_set_int @ B @ C4 ) )
& ~ ( member_set_int @ B @ C4 )
& ( B4
= ( insert_set_int @ A @ C4 ) )
& ~ ( member_set_int @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_937_insert__eq__iff,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B: nat > set_int,B4: set_nat_set_int] :
( ~ ( member_nat_set_int @ A @ A2 )
=> ( ~ ( member_nat_set_int @ B @ B4 )
=> ( ( ( insert_nat_set_int @ A @ A2 )
= ( insert_nat_set_int @ B @ B4 ) )
= ( ( ( A = B )
=> ( A2 = B4 ) )
& ( ( A != B )
=> ? [C4: set_nat_set_int] :
( ( A2
= ( insert_nat_set_int @ B @ C4 ) )
& ~ ( member_nat_set_int @ B @ C4 )
& ( B4
= ( insert_nat_set_int @ A @ C4 ) )
& ~ ( member_nat_set_int @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_938_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_939_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_940_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_941_insert__ident,axiom,
! [X: nat,A2: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B4 )
=> ( ( ( insert_nat @ X @ A2 )
= ( insert_nat @ X @ B4 ) )
= ( A2 = B4 ) ) ) ) ).
% insert_ident
thf(fact_942_insert__ident,axiom,
! [X: set_int,A2: set_set_int,B4: set_set_int] :
( ~ ( member_set_int @ X @ A2 )
=> ( ~ ( member_set_int @ X @ B4 )
=> ( ( ( insert_set_int @ X @ A2 )
= ( insert_set_int @ X @ B4 ) )
= ( A2 = B4 ) ) ) ) ).
% insert_ident
thf(fact_943_insert__ident,axiom,
! [X: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ~ ( member_nat_set_int @ X @ A2 )
=> ( ~ ( member_nat_set_int @ X @ B4 )
=> ( ( ( insert_nat_set_int @ X @ A2 )
= ( insert_nat_set_int @ X @ B4 ) )
= ( A2 = B4 ) ) ) ) ).
% insert_ident
thf(fact_944_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B5: set_nat] :
( ( A2
= ( insert_nat @ X @ B5 ) )
=> ( member_nat @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_945_Set_Oset__insert,axiom,
! [X: set_int,A2: set_set_int] :
( ( member_set_int @ X @ A2 )
=> ~ ! [B5: set_set_int] :
( ( A2
= ( insert_set_int @ X @ B5 ) )
=> ( member_set_int @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_946_Set_Oset__insert,axiom,
! [X: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ X @ A2 )
=> ~ ! [B5: set_nat_set_int] :
( ( A2
= ( insert_nat_set_int @ X @ B5 ) )
=> ( member_nat_set_int @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_947_insertI2,axiom,
! [A: nat,B4: set_nat,B: nat] :
( ( member_nat @ A @ B4 )
=> ( member_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).
% insertI2
thf(fact_948_insertI2,axiom,
! [A: set_int,B4: set_set_int,B: set_int] :
( ( member_set_int @ A @ B4 )
=> ( member_set_int @ A @ ( insert_set_int @ B @ B4 ) ) ) ).
% insertI2
thf(fact_949_insertI2,axiom,
! [A: nat > set_int,B4: set_nat_set_int,B: nat > set_int] :
( ( member_nat_set_int @ A @ B4 )
=> ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ B4 ) ) ) ).
% insertI2
thf(fact_950_insertI1,axiom,
! [A: nat,B4: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B4 ) ) ).
% insertI1
thf(fact_951_insertI1,axiom,
! [A: set_int,B4: set_set_int] : ( member_set_int @ A @ ( insert_set_int @ A @ B4 ) ) ).
% insertI1
thf(fact_952_insertI1,axiom,
! [A: nat > set_int,B4: set_nat_set_int] : ( member_nat_set_int @ A @ ( insert_nat_set_int @ A @ B4 ) ) ).
% insertI1
thf(fact_953_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_954_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_955_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_956_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_957_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_958_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_959_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_960_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_961_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_962_DiffD2,axiom,
! [C: nat,A2: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B4 ) )
=> ~ ( member_nat @ C @ B4 ) ) ).
% DiffD2
thf(fact_963_DiffD2,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B4 ) )
=> ~ ( member_nat_set_int @ C @ B4 ) ) ).
% DiffD2
thf(fact_964_DiffD2,axiom,
! [C: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A2 @ B4 ) )
=> ~ ( member_set_int @ C @ B4 ) ) ).
% DiffD2
thf(fact_965_DiffD1,axiom,
! [C: nat,A2: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B4 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_966_DiffD1,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B4 ) )
=> ( member_nat_set_int @ C @ A2 ) ) ).
% DiffD1
thf(fact_967_DiffD1,axiom,
! [C: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A2 @ B4 ) )
=> ( member_set_int @ C @ A2 ) ) ).
% DiffD1
thf(fact_968_DiffE,axiom,
! [C: nat,A2: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B4 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B4 ) ) ) ).
% DiffE
thf(fact_969_DiffE,axiom,
! [C: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ C @ ( minus_3247115583872269408et_int @ A2 @ B4 ) )
=> ~ ( ( member_nat_set_int @ C @ A2 )
=> ( member_nat_set_int @ C @ B4 ) ) ) ).
% DiffE
thf(fact_970_DiffE,axiom,
! [C: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A2 @ B4 ) )
=> ~ ( ( member_set_int @ C @ A2 )
=> ( member_set_int @ C @ B4 ) ) ) ).
% DiffE
thf(fact_971_psubsetD,axiom,
! [A2: set_nat,B4: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B4 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_972_psubsetD,axiom,
! [A2: set_set_int,B4: set_set_int,C: set_int] :
( ( ord_less_set_set_int @ A2 @ B4 )
=> ( ( member_set_int @ C @ A2 )
=> ( member_set_int @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_973_psubsetD,axiom,
! [A2: set_nat_set_int,B4: set_nat_set_int,C: nat > set_int] :
( ( ord_le2931775347370382171et_int @ A2 @ B4 )
=> ( ( member_nat_set_int @ C @ A2 )
=> ( member_nat_set_int @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_974_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_975_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_976_insert__mono,axiom,
! [C5: set_nat,D3: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C5 @ D3 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C5 ) @ ( insert_nat @ A @ D3 ) ) ) ).
% insert_mono
thf(fact_977_insert__mono,axiom,
! [C5: set_set_int,D3: set_set_int,A: set_int] :
( ( ord_le4403425263959731960et_int @ C5 @ D3 )
=> ( ord_le4403425263959731960et_int @ ( insert_set_int @ A @ C5 ) @ ( insert_set_int @ A @ D3 ) ) ) ).
% insert_mono
thf(fact_978_subset__insert,axiom,
! [X: nat,A2: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B4 ) )
= ( ord_less_eq_set_nat @ A2 @ B4 ) ) ) ).
% subset_insert
thf(fact_979_subset__insert,axiom,
! [X: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ~ ( member_nat_set_int @ X @ A2 )
=> ( ( ord_le5995675665013768039et_int @ A2 @ ( insert_nat_set_int @ X @ B4 ) )
= ( ord_le5995675665013768039et_int @ A2 @ B4 ) ) ) ).
% subset_insert
thf(fact_980_subset__insert,axiom,
! [X: set_int,A2: set_set_int,B4: set_set_int] :
( ~ ( member_set_int @ X @ A2 )
=> ( ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X @ B4 ) )
= ( ord_le4403425263959731960et_int @ A2 @ B4 ) ) ) ).
% subset_insert
thf(fact_981_subset__insertI,axiom,
! [B4: set_nat,A: nat] : ( ord_less_eq_set_nat @ B4 @ ( insert_nat @ A @ B4 ) ) ).
% subset_insertI
thf(fact_982_subset__insertI,axiom,
! [B4: set_set_int,A: set_int] : ( ord_le4403425263959731960et_int @ B4 @ ( insert_set_int @ A @ B4 ) ) ).
% subset_insertI
thf(fact_983_subset__insertI2,axiom,
! [A2: set_nat,B4: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_984_subset__insertI2,axiom,
! [A2: set_set_int,B4: set_set_int,B: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_985_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_986_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_987_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_988_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_989_doubleton__eq__iff,axiom,
! [A: set_int,B: set_int,C: set_int,D: set_int] :
( ( ( insert_set_int @ A @ ( insert_set_int @ B @ bot_bot_set_set_int ) )
= ( insert_set_int @ C @ ( insert_set_int @ D @ bot_bot_set_set_int ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_990_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_991_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_992_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_993_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_994_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_995_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_996_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_997_double__diff,axiom,
! [A2: set_set_int,B4: set_set_int,C5: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ( ord_le4403425263959731960et_int @ B4 @ C5 )
=> ( ( minus_8897228262479074673et_int @ B4 @ ( minus_8897228262479074673et_int @ C5 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_998_Diff__subset,axiom,
! [A2: set_set_int,B4: set_set_int] : ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A2 @ B4 ) @ A2 ) ).
% Diff_subset
thf(fact_999_Diff__mono,axiom,
! [A2: set_set_int,C5: set_set_int,D3: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ C5 )
=> ( ( ord_le4403425263959731960et_int @ D3 @ B4 )
=> ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A2 @ B4 ) @ ( minus_8897228262479074673et_int @ C5 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_1000_insert__Diff__if,axiom,
! [X: nat,B4: set_nat,A2: set_nat] :
( ( ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B4 )
= ( minus_minus_set_nat @ A2 @ B4 ) ) )
& ( ~ ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B4 )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1001_insert__Diff__if,axiom,
! [X: nat > set_int,B4: set_nat_set_int,A2: set_nat_set_int] :
( ( ( member_nat_set_int @ X @ B4 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X @ A2 ) @ B4 )
= ( minus_3247115583872269408et_int @ A2 @ B4 ) ) )
& ( ~ ( member_nat_set_int @ X @ B4 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X @ A2 ) @ B4 )
= ( insert_nat_set_int @ X @ ( minus_3247115583872269408et_int @ A2 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1002_insert__Diff__if,axiom,
! [X: set_int,B4: set_set_int,A2: set_set_int] :
( ( ( member_set_int @ X @ B4 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X @ A2 ) @ B4 )
= ( minus_8897228262479074673et_int @ A2 @ B4 ) ) )
& ( ~ ( member_set_int @ X @ B4 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X @ A2 ) @ B4 )
= ( insert_set_int @ X @ ( minus_8897228262479074673et_int @ A2 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1003_psubsetE,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B4 )
=> ~ ( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ord_le4403425263959731960et_int @ B4 @ A2 ) ) ) ).
% psubsetE
thf(fact_1004_psubset__eq,axiom,
( ord_less_set_set_int
= ( ^ [A5: set_set_int,B6: set_set_int] :
( ( ord_le4403425263959731960et_int @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_1005_psubset__imp__subset,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B4 )
=> ( ord_le4403425263959731960et_int @ A2 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_1006_psubset__subset__trans,axiom,
! [A2: set_set_int,B4: set_set_int,C5: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B4 )
=> ( ( ord_le4403425263959731960et_int @ B4 @ C5 )
=> ( ord_less_set_set_int @ A2 @ C5 ) ) ) ).
% psubset_subset_trans
thf(fact_1007_subset__not__subset__eq,axiom,
( ord_less_set_set_int
= ( ^ [A5: set_set_int,B6: set_set_int] :
( ( ord_le4403425263959731960et_int @ A5 @ B6 )
& ~ ( ord_le4403425263959731960et_int @ B6 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1008_subset__psubset__trans,axiom,
! [A2: set_set_int,B4: set_set_int,C5: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B4 )
=> ( ( ord_less_set_set_int @ B4 @ C5 )
=> ( ord_less_set_set_int @ A2 @ C5 ) ) ) ).
% subset_psubset_trans
thf(fact_1009_subset__iff__psubset__eq,axiom,
( ord_le4403425263959731960et_int
= ( ^ [A5: set_set_int,B6: set_set_int] :
( ( ord_less_set_set_int @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1010_not__psubset__empty,axiom,
! [A2: set_set_int] :
~ ( ord_less_set_set_int @ A2 @ bot_bot_set_set_int ) ).
% not_psubset_empty
thf(fact_1011_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_1012_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A2 @ B4 )
=> ? [B2: nat] : ( member_nat @ B2 @ ( minus_minus_set_nat @ B4 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1013_psubset__imp__ex__mem,axiom,
! [A2: set_nat_set_int,B4: set_nat_set_int] :
( ( ord_le2931775347370382171et_int @ A2 @ B4 )
=> ? [B2: nat > set_int] : ( member_nat_set_int @ B2 @ ( minus_3247115583872269408et_int @ B4 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1014_psubset__imp__ex__mem,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B4 )
=> ? [B2: set_int] : ( member_set_int @ B2 @ ( minus_8897228262479074673et_int @ B4 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1015_subset__singleton__iff,axiom,
! [X5: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1016_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_1017_subset__singletonD,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1018_subset__singletonD,axiom,
! [A2: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X @ bot_bot_set_set_int ) )
=> ( ( A2 = bot_bot_set_set_int )
| ( A2
= ( insert_set_int @ X @ bot_bot_set_set_int ) ) ) ) ).
% subset_singletonD
thf(fact_1019_subset__Diff__insert,axiom,
! [A2: set_nat,B4: set_nat,X: nat,C5: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B4 @ ( insert_nat @ X @ C5 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B4 @ C5 ) )
& ~ ( member_nat @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1020_subset__Diff__insert,axiom,
! [A2: set_nat_set_int,B4: set_nat_set_int,X: nat > set_int,C5: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ ( minus_3247115583872269408et_int @ B4 @ ( insert_nat_set_int @ X @ C5 ) ) )
= ( ( ord_le5995675665013768039et_int @ A2 @ ( minus_3247115583872269408et_int @ B4 @ C5 ) )
& ~ ( member_nat_set_int @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1021_subset__Diff__insert,axiom,
! [A2: set_set_int,B4: set_set_int,X: set_int,C5: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( minus_8897228262479074673et_int @ B4 @ ( insert_set_int @ X @ C5 ) ) )
= ( ( ord_le4403425263959731960et_int @ A2 @ ( minus_8897228262479074673et_int @ B4 @ C5 ) )
& ~ ( member_set_int @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1022_Diff__insert__absorb,axiom,
! [X: nat > set_int,A2: set_nat_set_int] :
( ~ ( member_nat_set_int @ X @ A2 )
=> ( ( minus_3247115583872269408et_int @ ( insert_nat_set_int @ X @ A2 ) @ ( insert_nat_set_int @ X @ bot_bo8417611410066262939et_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1023_Diff__insert__absorb,axiom,
! [X: nat,A2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1024_Diff__insert__absorb,axiom,
! [X: set_int,A2: set_set_int] :
( ~ ( member_set_int @ X @ A2 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X @ A2 ) @ ( insert_set_int @ X @ bot_bot_set_set_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1025_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B4: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B4 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_1026_Diff__insert2,axiom,
! [A2: set_set_int,A: set_int,B4: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ B4 ) )
= ( minus_8897228262479074673et_int @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_1027_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_1028_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_1029_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_1030_Diff__insert,axiom,
! [A2: set_nat,A: nat,B4: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B4 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B4 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_1031_Diff__insert,axiom,
! [A2: set_set_int,A: set_int,B4: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ B4 ) )
= ( minus_8897228262479074673et_int @ ( minus_8897228262479074673et_int @ A2 @ B4 ) @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ) ).
% Diff_insert
thf(fact_1032_Diff__single__insert,axiom,
! [A2: set_nat,X: nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_1033_Diff__single__insert,axiom,
! [A2: set_set_int,X: set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ X @ bot_bot_set_set_int ) ) @ B4 )
=> ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_1034_subset__insert__iff,axiom,
! [A2: set_nat_set_int,X: nat > set_int,B4: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ ( insert_nat_set_int @ X @ B4 ) )
= ( ( ( member_nat_set_int @ X @ A2 )
=> ( ord_le5995675665013768039et_int @ ( minus_3247115583872269408et_int @ A2 @ ( insert_nat_set_int @ X @ bot_bo8417611410066262939et_int ) ) @ B4 ) )
& ( ~ ( member_nat_set_int @ X @ A2 )
=> ( ord_le5995675665013768039et_int @ A2 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_1035_subset__insert__iff,axiom,
! [A2: set_nat,X: nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B4 ) )
= ( ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_1036_subset__insert__iff,axiom,
! [A2: set_set_int,X: set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X @ B4 ) )
= ( ( ( member_set_int @ X @ A2 )
=> ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ X @ bot_bot_set_set_int ) ) @ B4 ) )
& ( ~ ( member_set_int @ X @ A2 )
=> ( ord_le4403425263959731960et_int @ A2 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_1037_psubset__insert__iff,axiom,
! [A2: set_nat_set_int,X: nat > set_int,B4: set_nat_set_int] :
( ( ord_le2931775347370382171et_int @ A2 @ ( insert_nat_set_int @ X @ B4 ) )
= ( ( ( member_nat_set_int @ X @ B4 )
=> ( ord_le2931775347370382171et_int @ A2 @ B4 ) )
& ( ~ ( member_nat_set_int @ X @ B4 )
=> ( ( ( member_nat_set_int @ X @ A2 )
=> ( ord_le2931775347370382171et_int @ ( minus_3247115583872269408et_int @ A2 @ ( insert_nat_set_int @ X @ bot_bo8417611410066262939et_int ) ) @ B4 ) )
& ( ~ ( member_nat_set_int @ X @ A2 )
=> ( ord_le5995675665013768039et_int @ A2 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1038_psubset__insert__iff,axiom,
! [A2: set_nat,X: nat,B4: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B4 ) )
= ( ( ( member_nat @ X @ B4 )
=> ( ord_less_set_nat @ A2 @ B4 ) )
& ( ~ ( member_nat @ X @ B4 )
=> ( ( ( member_nat @ X @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1039_psubset__insert__iff,axiom,
! [A2: set_set_int,X: set_int,B4: set_set_int] :
( ( ord_less_set_set_int @ A2 @ ( insert_set_int @ X @ B4 ) )
= ( ( ( member_set_int @ X @ B4 )
=> ( ord_less_set_set_int @ A2 @ B4 ) )
& ( ~ ( member_set_int @ X @ B4 )
=> ( ( ( member_set_int @ X @ A2 )
=> ( ord_less_set_set_int @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ X @ bot_bot_set_set_int ) ) @ B4 ) )
& ( ~ ( member_set_int @ X @ A2 )
=> ( ord_le4403425263959731960et_int @ A2 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1040_r_Oa__lcos__m__assoc,axiom,
! [M6: set_set_int,G2: set_int,H: set_int] :
( ( ord_le4403425263959731960et_int @ M6 @ ( 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 @ M6 ) )
= ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ G2 @ H ) @ M6 ) ) ) ) ) ).
% r.a_lcos_m_assoc
thf(fact_1041_r_Oa__lcos__mult__one,axiom,
! [M6: set_set_int] :
( ( ord_le4403425263959731960et_int @ M6 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ M6 )
= M6 ) ) ).
% r.a_lcos_mult_one
thf(fact_1042_r_Oset__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 ) ) ).
% r.set_add_zero
thf(fact_1043_r_Oset__add__closed,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ B4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A2 @ B4 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.set_add_closed
thf(fact_1044_r_Oset__add__comm,axiom,
! [I4: set_set_int,J4: set_set_int] :
( ( ord_le4403425263959731960et_int @ I4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ J4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I4 @ J4 )
= ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ J4 @ I4 ) ) ) ) ).
% r.set_add_comm
thf(fact_1045_r_Osetadd__subset__G,axiom,
! [H2: set_set_int,K2: set_set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 @ K2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.setadd_subset_G
thf(fact_1046_r_Oa__l__coset__subset__G,axiom,
! [H2: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ H2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.a_l_coset_subset_G
thf(fact_1047_r_Oadd__additive__subgroups,axiom,
! [H2: set_set_int,K2: set_set_int] :
( ( additi7073586575563672860t_unit @ H2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( additi7073586575563672860t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( additi7073586575563672860t_unit @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 @ K2 ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.add_additive_subgroups
thf(fact_1048_r_Ospace__subgroup__props_I6_J,axiom,
! [K2: set_set_int,N: nat,E: set_set_int,K: set_int,A: set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K2 @ E )
=> ( ( member_set_int @ K @ ( minus_8897228262479074673et_int @ K2 @ ( 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 ) ) @ K @ A ) @ E )
=> ( member_set_int @ A @ E ) ) ) ) ) ) ).
% r.space_subgroup_props(6)
thf(fact_1049_r_Osubring__props_I2_J,axiom,
! [K2: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ K2 ) ) ).
% r.subring_props(2)
thf(fact_1050_r_Osubring__props_I7_J,axiom,
! [K2: set_set_int,H1: set_int,H22: set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ H1 @ K2 )
=> ( ( member_set_int @ H22 @ K2 )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H1 @ H22 ) @ K2 ) ) ) ) ).
% r.subring_props(7)
thf(fact_1051_r_Osubring__props_I6_J,axiom,
! [K2: set_set_int,H1: set_int,H22: set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ H1 @ K2 )
=> ( ( member_set_int @ H22 @ K2 )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H1 @ H22 ) @ K2 ) ) ) ) ).
% r.subring_props(6)
thf(fact_1052_r_Osubring__props_I4_J,axiom,
! [K2: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( K2 != bot_bot_set_set_int ) ) ).
% r.subring_props(4)
thf(fact_1053_r_Odimension__is__inj,axiom,
! [K2: set_set_int,N: nat,E: set_set_int,M: nat] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K2 @ E )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ K2 @ E )
=> ( N = M ) ) ) ) ).
% r.dimension_is_inj
thf(fact_1054_r_Osubring__props_I1_J,axiom,
! [K2: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ord_le4403425263959731960et_int @ K2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.subring_props(1)
thf(fact_1055_r_Ospace__subgroup__props_I2_J,axiom,
! [K2: set_set_int,N: nat,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K2 @ E )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ E ) ) ) ).
% r.space_subgroup_props(2)
thf(fact_1056_r_Ospace__subgroup__props_I3_J,axiom,
! [K2: set_set_int,N: nat,E: set_set_int,V1: set_int,V22: set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K2 @ E )
=> ( ( member_set_int @ V1 @ E )
=> ( ( member_set_int @ V22 @ E )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ V1 @ V22 ) @ E ) ) ) ) ) ).
% r.space_subgroup_props(3)
thf(fact_1057_r_Otelescopic__base__aux,axiom,
! [K2: set_set_int,F4: set_set_int,N: nat,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( subfie3888952257595785920t_unit @ F4 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K2 @ F4 )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ one_one_nat @ F4 @ E )
=> ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K2 @ E ) ) ) ) ) ).
% r.telescopic_base_aux
thf(fact_1058_r_Otelescopic__base,axiom,
! [K2: set_set_int,F4: set_set_int,N: nat,M: nat,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( subfie3888952257595785920t_unit @ F4 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K2 @ F4 )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ F4 @ E )
=> ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( times_times_nat @ N @ M ) @ K2 @ E ) ) ) ) ) ).
% r.telescopic_base
thf(fact_1059_r_Ospace__subgroup__props_I5_J,axiom,
! [K2: set_set_int,N: nat,E: set_set_int,K: set_int,V4: set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K2 @ E )
=> ( ( member_set_int @ K @ K2 )
=> ( ( member_set_int @ V4 @ E )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ V4 ) @ E ) ) ) ) ) ).
% r.space_subgroup_props(5)
thf(fact_1060_r_Ospace__subgroup__props_I1_J,axiom,
! [K2: set_set_int,N: nat,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K2 @ E )
=> ( ord_le4403425263959731960et_int @ E @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.space_subgroup_props(1)
thf(fact_1061_r_Oline__extension__smult__closed,axiom,
! [K2: set_set_int,E: set_set_int,A: set_int,K: set_int,U: set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ! [K4: set_int,V2: set_int] :
( ( member_set_int @ K4 @ K2 )
=> ( ( member_set_int @ V2 @ E )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K4 @ 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 @ K @ K2 )
=> ( ( member_set_int @ U @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ A @ E ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ U ) @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ A @ E ) ) ) ) ) ) ) ) ).
% r.line_extension_smult_closed
thf(fact_1062_r_Odimension__backwards,axiom,
! [K2: set_set_int,N: nat,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( suc @ N ) @ K2 @ E )
=> ? [X3: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ? [E5: set_set_int] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K2 @ E5 )
& ~ ( member_set_int @ X3 @ E5 )
& ( E
= ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ X3 @ E5 ) ) ) ) ) ) ).
% r.dimension_backwards
thf(fact_1063_r_Odimension__zero,axiom,
! [K2: set_set_int,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ zero_zero_nat @ K2 @ E )
=> ( E
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) ) ) ).
% r.dimension_zero
thf(fact_1064_r_Osubfield__m__inv__simprule,axiom,
! [K2: set_set_int,K: set_int,A: set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ K @ ( minus_8897228262479074673et_int @ K2 @ ( 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 ) ) @ K @ A ) @ K2 )
=> ( member_set_int @ A @ K2 ) ) ) ) ) ).
% r.subfield_m_inv_simprule
thf(fact_1065_r_Odimension__one,axiom,
! [K2: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ one_one_nat @ K2 @ K2 ) ) ).
% r.dimension_one
thf(fact_1066_r_Odimension__direct__sum__space,axiom,
! [K2: set_set_int,N: nat,E: set_set_int,M: nat,F4: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K2 @ E )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ K2 @ F4 )
=> ( ( ( inf_inf_set_set_int @ E @ F4 )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
=> ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( plus_plus_nat @ N @ M ) @ K2 @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ E @ F4 ) ) ) ) ) ) ).
% r.dimension_direct_sum_space
thf(fact_1067_r_Odimension__sum__space,axiom,
! [K2: set_set_int,N: nat,E: set_set_int,M: nat,F4: set_set_int,K: nat] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K2 @ E )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ K2 @ F4 )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ K2 @ ( inf_inf_set_set_int @ E @ F4 ) )
=> ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ K ) @ K2 @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ E @ F4 ) ) ) ) ) ) ).
% r.dimension_sum_space
thf(fact_1068_r_Osubcring__inter,axiom,
! [I4: set_set_int,J4: set_set_int] :
( ( subcri1024317279029940167t_unit @ I4 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( subcri1024317279029940167t_unit @ J4 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( subcri1024317279029940167t_unit @ ( inf_inf_set_set_int @ I4 @ J4 ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.subcring_inter
thf(fact_1069_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B4: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B4 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1070_Int__insert__right__if1,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ A @ A2 )
=> ( ( inf_in1752217752563533465et_int @ A2 @ ( insert_nat_set_int @ A @ B4 ) )
= ( insert_nat_set_int @ A @ ( inf_in1752217752563533465et_int @ A2 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1071_Int__insert__right__if1,axiom,
! [A: set_int,A2: set_set_int,B4: set_set_int] :
( ( member_set_int @ A @ A2 )
=> ( ( inf_inf_set_set_int @ A2 @ ( insert_set_int @ A @ B4 ) )
= ( insert_set_int @ A @ ( inf_inf_set_set_int @ A2 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1072_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B4: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B4 ) )
= ( inf_inf_set_nat @ A2 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_1073_Int__insert__right__if0,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ~ ( member_nat_set_int @ A @ A2 )
=> ( ( inf_in1752217752563533465et_int @ A2 @ ( insert_nat_set_int @ A @ B4 ) )
= ( inf_in1752217752563533465et_int @ A2 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_1074_Int__insert__right__if0,axiom,
! [A: set_int,A2: set_set_int,B4: set_set_int] :
( ~ ( member_set_int @ A @ A2 )
=> ( ( inf_inf_set_set_int @ A2 @ ( insert_set_int @ A @ B4 ) )
= ( inf_inf_set_set_int @ A2 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_1075_insert__inter__insert,axiom,
! [A: nat,A2: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ ( insert_nat @ A @ B4 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_1076_insert__inter__insert,axiom,
! [A: set_int,A2: set_set_int,B4: set_set_int] :
( ( inf_inf_set_set_int @ ( insert_set_int @ A @ A2 ) @ ( insert_set_int @ A @ B4 ) )
= ( insert_set_int @ A @ ( inf_inf_set_set_int @ A2 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_1077_Int__insert__left__if1,axiom,
! [A: nat,C5: set_nat,B4: set_nat] :
( ( member_nat @ A @ C5 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B4 ) @ C5 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B4 @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1078_Int__insert__left__if1,axiom,
! [A: nat > set_int,C5: set_nat_set_int,B4: set_nat_set_int] :
( ( member_nat_set_int @ A @ C5 )
=> ( ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ B4 ) @ C5 )
= ( insert_nat_set_int @ A @ ( inf_in1752217752563533465et_int @ B4 @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1079_Int__insert__left__if1,axiom,
! [A: set_int,C5: set_set_int,B4: set_set_int] :
( ( member_set_int @ A @ C5 )
=> ( ( inf_inf_set_set_int @ ( insert_set_int @ A @ B4 ) @ C5 )
= ( insert_set_int @ A @ ( inf_inf_set_set_int @ B4 @ C5 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1080_Int__insert__left__if0,axiom,
! [A: nat,C5: set_nat,B4: set_nat] :
( ~ ( member_nat @ A @ C5 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B4 ) @ C5 )
= ( inf_inf_set_nat @ B4 @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_1081_Int__insert__left__if0,axiom,
! [A: nat > set_int,C5: set_nat_set_int,B4: set_nat_set_int] :
( ~ ( member_nat_set_int @ A @ C5 )
=> ( ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ B4 ) @ C5 )
= ( inf_in1752217752563533465et_int @ B4 @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_1082_Int__insert__left__if0,axiom,
! [A: set_int,C5: set_set_int,B4: set_set_int] :
( ~ ( member_set_int @ A @ C5 )
=> ( ( inf_inf_set_set_int @ ( insert_set_int @ A @ B4 ) @ C5 )
= ( inf_inf_set_set_int @ B4 @ C5 ) ) ) ).
% Int_insert_left_if0
thf(fact_1083_disjoint__insert_I2_J,axiom,
! [A2: set_nat_set_int,B: nat > set_int,B4: set_nat_set_int] :
( ( bot_bo8417611410066262939et_int
= ( inf_in1752217752563533465et_int @ A2 @ ( insert_nat_set_int @ B @ B4 ) ) )
= ( ~ ( member_nat_set_int @ B @ A2 )
& ( bot_bo8417611410066262939et_int
= ( inf_in1752217752563533465et_int @ A2 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1084_disjoint__insert_I2_J,axiom,
! [A2: set_set_int,B: set_int,B4: set_set_int] :
( ( bot_bot_set_set_int
= ( inf_inf_set_set_int @ A2 @ ( insert_set_int @ B @ B4 ) ) )
= ( ~ ( member_set_int @ B @ A2 )
& ( bot_bot_set_set_int
= ( inf_inf_set_set_int @ A2 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1085_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B: nat,B4: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat @ B @ B4 ) ) )
= ( ~ ( member_nat @ B @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1086_disjoint__insert_I1_J,axiom,
! [B4: set_nat_set_int,A: nat > set_int,A2: set_nat_set_int] :
( ( ( inf_in1752217752563533465et_int @ B4 @ ( insert_nat_set_int @ A @ A2 ) )
= bot_bo8417611410066262939et_int )
= ( ~ ( member_nat_set_int @ A @ B4 )
& ( ( inf_in1752217752563533465et_int @ B4 @ A2 )
= bot_bo8417611410066262939et_int ) ) ) ).
% disjoint_insert(1)
thf(fact_1087_disjoint__insert_I1_J,axiom,
! [B4: set_set_int,A: set_int,A2: set_set_int] :
( ( ( inf_inf_set_set_int @ B4 @ ( insert_set_int @ A @ A2 ) )
= bot_bot_set_set_int )
= ( ~ ( member_set_int @ A @ B4 )
& ( ( inf_inf_set_set_int @ B4 @ A2 )
= bot_bot_set_set_int ) ) ) ).
% disjoint_insert(1)
thf(fact_1088_disjoint__insert_I1_J,axiom,
! [B4: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B4 @ ( insert_nat @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B4 )
& ( ( inf_inf_set_nat @ B4 @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_1089_insert__disjoint_I2_J,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( bot_bo8417611410066262939et_int
= ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ A2 ) @ B4 ) )
= ( ~ ( member_nat_set_int @ A @ B4 )
& ( bot_bo8417611410066262939et_int
= ( inf_in1752217752563533465et_int @ A2 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1090_insert__disjoint_I2_J,axiom,
! [A: set_int,A2: set_set_int,B4: set_set_int] :
( ( bot_bot_set_set_int
= ( inf_inf_set_set_int @ ( insert_set_int @ A @ A2 ) @ B4 ) )
= ( ~ ( member_set_int @ A @ B4 )
& ( bot_bot_set_set_int
= ( inf_inf_set_set_int @ A2 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1091_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B4: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B4 ) )
= ( ~ ( member_nat @ A @ B4 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1092_insert__disjoint_I1_J,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ A2 ) @ B4 )
= bot_bo8417611410066262939et_int )
= ( ~ ( member_nat_set_int @ A @ B4 )
& ( ( inf_in1752217752563533465et_int @ A2 @ B4 )
= bot_bo8417611410066262939et_int ) ) ) ).
% insert_disjoint(1)
thf(fact_1093_insert__disjoint_I1_J,axiom,
! [A: set_int,A2: set_set_int,B4: set_set_int] :
( ( ( inf_inf_set_set_int @ ( insert_set_int @ A @ A2 ) @ B4 )
= bot_bot_set_set_int )
= ( ~ ( member_set_int @ A @ B4 )
& ( ( inf_inf_set_set_int @ A2 @ B4 )
= bot_bot_set_set_int ) ) ) ).
% insert_disjoint(1)
thf(fact_1094_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B4 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B4 )
& ( ( inf_inf_set_nat @ A2 @ B4 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_1095_Diff__disjoint,axiom,
! [A2: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B4 @ A2 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_1096_Diff__disjoint,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( inf_inf_set_set_int @ A2 @ ( minus_8897228262479074673et_int @ B4 @ A2 ) )
= bot_bot_set_set_int ) ).
% Diff_disjoint
thf(fact_1097_Diff__triv,axiom,
! [A2: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B4 )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A2 @ B4 )
= A2 ) ) ).
% Diff_triv
thf(fact_1098_Diff__triv,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ( inf_inf_set_set_int @ A2 @ B4 )
= bot_bot_set_set_int )
=> ( ( minus_8897228262479074673et_int @ A2 @ B4 )
= A2 ) ) ).
% Diff_triv
thf(fact_1099_Int__Diff__disjoint,axiom,
! [A2: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B4 ) @ ( minus_minus_set_nat @ A2 @ B4 ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_1100_Int__Diff__disjoint,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( inf_inf_set_set_int @ ( inf_inf_set_set_int @ A2 @ B4 ) @ ( minus_8897228262479074673et_int @ A2 @ B4 ) )
= bot_bot_set_set_int ) ).
% Int_Diff_disjoint
thf(fact_1101_Diff__Int__distrib2,axiom,
! [A2: set_set_int,B4: set_set_int,C5: set_set_int] :
( ( inf_inf_set_set_int @ ( minus_8897228262479074673et_int @ A2 @ B4 ) @ C5 )
= ( minus_8897228262479074673et_int @ ( inf_inf_set_set_int @ A2 @ C5 ) @ ( inf_inf_set_set_int @ B4 @ C5 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1102_Diff__Int__distrib,axiom,
! [C5: set_set_int,A2: set_set_int,B4: set_set_int] :
( ( inf_inf_set_set_int @ C5 @ ( minus_8897228262479074673et_int @ A2 @ B4 ) )
= ( minus_8897228262479074673et_int @ ( inf_inf_set_set_int @ C5 @ A2 ) @ ( inf_inf_set_set_int @ C5 @ B4 ) ) ) ).
% Diff_Int_distrib
thf(fact_1103_Diff__Diff__Int,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ ( minus_8897228262479074673et_int @ A2 @ B4 ) )
= ( inf_inf_set_set_int @ A2 @ B4 ) ) ).
% Diff_Diff_Int
thf(fact_1104_Diff__Int2,axiom,
! [A2: set_set_int,C5: set_set_int,B4: set_set_int] :
( ( minus_8897228262479074673et_int @ ( inf_inf_set_set_int @ A2 @ C5 ) @ ( inf_inf_set_set_int @ B4 @ C5 ) )
= ( minus_8897228262479074673et_int @ ( inf_inf_set_set_int @ A2 @ C5 ) @ B4 ) ) ).
% Diff_Int2
thf(fact_1105_Int__Diff,axiom,
! [A2: set_set_int,B4: set_set_int,C5: set_set_int] :
( ( minus_8897228262479074673et_int @ ( inf_inf_set_set_int @ A2 @ B4 ) @ C5 )
= ( inf_inf_set_set_int @ A2 @ ( minus_8897228262479074673et_int @ B4 @ C5 ) ) ) ).
% Int_Diff
thf(fact_1106_Int__insert__right,axiom,
! [A: nat,A2: set_nat,B4: set_nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B4 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B4 ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B4 ) )
= ( inf_inf_set_nat @ A2 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1107_Int__insert__right,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B4: set_nat_set_int] :
( ( ( member_nat_set_int @ A @ A2 )
=> ( ( inf_in1752217752563533465et_int @ A2 @ ( insert_nat_set_int @ A @ B4 ) )
= ( insert_nat_set_int @ A @ ( inf_in1752217752563533465et_int @ A2 @ B4 ) ) ) )
& ( ~ ( member_nat_set_int @ A @ A2 )
=> ( ( inf_in1752217752563533465et_int @ A2 @ ( insert_nat_set_int @ A @ B4 ) )
= ( inf_in1752217752563533465et_int @ A2 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1108_Int__insert__right,axiom,
! [A: set_int,A2: set_set_int,B4: set_set_int] :
( ( ( member_set_int @ A @ A2 )
=> ( ( inf_inf_set_set_int @ A2 @ ( insert_set_int @ A @ B4 ) )
= ( insert_set_int @ A @ ( inf_inf_set_set_int @ A2 @ B4 ) ) ) )
& ( ~ ( member_set_int @ A @ A2 )
=> ( ( inf_inf_set_set_int @ A2 @ ( insert_set_int @ A @ B4 ) )
= ( inf_inf_set_set_int @ A2 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1109_Int__insert__left,axiom,
! [A: nat,C5: set_nat,B4: set_nat] :
( ( ( member_nat @ A @ C5 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B4 ) @ C5 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B4 @ C5 ) ) ) )
& ( ~ ( member_nat @ A @ C5 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B4 ) @ C5 )
= ( inf_inf_set_nat @ B4 @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_1110_Int__insert__left,axiom,
! [A: nat > set_int,C5: set_nat_set_int,B4: set_nat_set_int] :
( ( ( member_nat_set_int @ A @ C5 )
=> ( ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ B4 ) @ C5 )
= ( insert_nat_set_int @ A @ ( inf_in1752217752563533465et_int @ B4 @ C5 ) ) ) )
& ( ~ ( member_nat_set_int @ A @ C5 )
=> ( ( inf_in1752217752563533465et_int @ ( insert_nat_set_int @ A @ B4 ) @ C5 )
= ( inf_in1752217752563533465et_int @ B4 @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_1111_Int__insert__left,axiom,
! [A: set_int,C5: set_set_int,B4: set_set_int] :
( ( ( member_set_int @ A @ C5 )
=> ( ( inf_inf_set_set_int @ ( insert_set_int @ A @ B4 ) @ C5 )
= ( insert_set_int @ A @ ( inf_inf_set_set_int @ B4 @ C5 ) ) ) )
& ( ~ ( member_set_int @ A @ C5 )
=> ( ( inf_inf_set_set_int @ ( insert_set_int @ A @ B4 ) @ C5 )
= ( inf_inf_set_set_int @ B4 @ C5 ) ) ) ) ).
% Int_insert_left
thf(fact_1112_disjoint__iff__not__equal,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ( inf_inf_set_set_int @ A2 @ B4 )
= bot_bot_set_set_int )
= ( ! [X2: set_int] :
( ( member_set_int @ X2 @ A2 )
=> ! [Y3: set_int] :
( ( member_set_int @ Y3 @ B4 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1113_disjoint__iff__not__equal,axiom,
! [A2: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B4 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ B4 )
=> ( X2 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1114_Int__empty__right,axiom,
! [A2: set_set_int] :
( ( inf_inf_set_set_int @ A2 @ bot_bot_set_set_int )
= bot_bot_set_set_int ) ).
% Int_empty_right
thf(fact_1115_Int__empty__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_1116_Int__empty__left,axiom,
! [B4: set_set_int] :
( ( inf_inf_set_set_int @ bot_bot_set_set_int @ B4 )
= bot_bot_set_set_int ) ).
% Int_empty_left
thf(fact_1117_Int__empty__left,axiom,
! [B4: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B4 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_1118_disjoint__iff,axiom,
! [A2: set_nat_set_int,B4: set_nat_set_int] :
( ( ( inf_in1752217752563533465et_int @ A2 @ B4 )
= bot_bo8417611410066262939et_int )
= ( ! [X2: nat > set_int] :
( ( member_nat_set_int @ X2 @ A2 )
=> ~ ( member_nat_set_int @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1119_disjoint__iff,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ( ( inf_inf_set_set_int @ A2 @ B4 )
= bot_bot_set_set_int )
= ( ! [X2: set_int] :
( ( member_set_int @ X2 @ A2 )
=> ~ ( member_set_int @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1120_disjoint__iff,axiom,
! [A2: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B4 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ X2 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_1121_Int__emptyI,axiom,
! [A2: set_nat_set_int,B4: set_nat_set_int] :
( ! [X3: nat > set_int] :
( ( member_nat_set_int @ X3 @ A2 )
=> ~ ( member_nat_set_int @ X3 @ B4 ) )
=> ( ( inf_in1752217752563533465et_int @ A2 @ B4 )
= bot_bo8417611410066262939et_int ) ) ).
% Int_emptyI
thf(fact_1122_Int__emptyI,axiom,
! [A2: set_set_int,B4: set_set_int] :
( ! [X3: set_int] :
( ( member_set_int @ X3 @ A2 )
=> ~ ( member_set_int @ X3 @ B4 ) )
=> ( ( inf_inf_set_set_int @ A2 @ B4 )
= bot_bot_set_set_int ) ) ).
% Int_emptyI
thf(fact_1123_Int__emptyI,axiom,
! [A2: set_nat,B4: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ X3 @ B4 ) )
=> ( ( inf_inf_set_nat @ A2 @ B4 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_1124_Iio__Int__singleton,axiom,
! [X: set_int,K: set_int] :
( ( ( ord_less_set_int @ X @ K )
=> ( ( inf_inf_set_set_int @ ( set_or5935648273017318783et_int @ K ) @ ( insert_set_int @ X @ bot_bot_set_set_int ) )
= ( insert_set_int @ X @ bot_bot_set_set_int ) ) )
& ( ~ ( ord_less_set_int @ X @ K )
=> ( ( inf_inf_set_set_int @ ( set_or5935648273017318783et_int @ K ) @ ( insert_set_int @ X @ bot_bot_set_set_int ) )
= bot_bot_set_set_int ) ) ) ).
% Iio_Int_singleton
thf(fact_1125_Iio__Int__singleton,axiom,
! [X: nat,K: nat] :
( ( ( ord_less_nat @ X @ K )
=> ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X @ K )
=> ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ) ) ).
% Iio_Int_singleton
thf(fact_1126_r_Ocarrier__is__subalgebra,axiom,
! [K2: set_set_int] :
( ( ord_le4403425263959731960et_int @ K2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( embedd2743979684206749024t_unit @ K2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.carrier_is_subalgebra
thf(fact_1127_r_Osubalgebra__in__carrier,axiom,
! [K2: set_set_int,V5: set_set_int] :
( ( embedd2743979684206749024t_unit @ K2 @ V5 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ord_le4403425263959731960et_int @ V5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.subalgebra_in_carrier
thf(fact_1128_r_Osubalgebra__inter,axiom,
! [K2: set_set_int,V5: set_set_int,V6: set_set_int] :
( ( embedd2743979684206749024t_unit @ K2 @ V5 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd2743979684206749024t_unit @ K2 @ V6 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( embedd2743979684206749024t_unit @ K2 @ ( inf_inf_set_set_int @ V5 @ V6 ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.subalgebra_inter
thf(fact_1129_r_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K2: set_set_int,E: set_set_int,V5: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ E )
=> ( ( embedd2743979684206749024t_unit @ K2 @ V5 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ord_le4403425263959731960et_int @ V5 @ E )
=> ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ V5 ) ) ) ) ) ).
% r.subalbegra_incl_imp_finite_dimension
thf(fact_1130_r_Ofinite__dimension__imp__subalgebra,axiom,
! [K2: set_set_int,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ E )
=> ( embedd2743979684206749024t_unit @ K2 @ E @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.finite_dimension_imp_subalgebra
thf(fact_1131_r_Otelescopic__base__dim_I1_J,axiom,
! [K2: set_set_int,F4: set_set_int,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( subfie3888952257595785920t_unit @ F4 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ F4 )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ F4 @ E )
=> ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ E ) ) ) ) ) ).
% r.telescopic_base_dim(1)
thf(fact_1132_r_Ofinite__dimensionE_H,axiom,
! [K2: set_set_int,E: set_set_int] :
( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ E )
=> ~ ! [N2: nat] :
~ ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N2 @ K2 @ E ) ) ).
% r.finite_dimensionE'
thf(fact_1133_r_Ofinite__dimensionI,axiom,
! [N: nat,K2: set_set_int,E: set_set_int] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K2 @ E )
=> ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ E ) ) ).
% r.finite_dimensionI
thf(fact_1134_r_Ofinite__dimension__def,axiom,
! [K2: set_set_int,E: set_set_int] :
( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ E )
= ( ? [N3: nat] : ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N3 @ K2 @ E ) ) ) ).
% r.finite_dimension_def
thf(fact_1135_r_Ounique__dimension,axiom,
! [K2: set_set_int,E: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ E )
=> ? [X3: nat] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X3 @ K2 @ E )
& ! [Y6: nat] :
( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y6 @ K2 @ E )
=> ( Y6 = X3 ) ) ) ) ) ).
% r.unique_dimension
thf(fact_1136_r_Osum__space__dim_I1_J,axiom,
! [K2: set_set_int,E: set_set_int,F4: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ E )
=> ( ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ F4 )
=> ( embedd8246663962306818995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ E @ F4 ) ) ) ) ) ).
% r.sum_space_dim(1)
thf(fact_1137_r_Ogenideal__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 ) ) ) ) ).
% r.genideal_one
thf(fact_1138_r_Osubring__props_I3_J,axiom,
! [K2: set_set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( member_set_int @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ K2 ) ) ).
% r.subring_props(3)
thf(fact_1139_r_Oone__unique,axiom,
! [U: set_int] :
( ( member_set_int @ U @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ! [X3: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ U @ X3 )
= X3 ) )
=> ( U
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.one_unique
thf(fact_1140_r_Oinv__unique,axiom,
! [Y: set_int,X: set_int,Y2: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y2 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( 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 ) ) ) ) ) ) ).
% r.inv_unique
thf(fact_1141_r_Oone__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 ) ) ) ) ) ).
% r.one_zeroI
thf(fact_1142_r_Oone__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 ) ) ) ).
% r.one_zeroD
thf(fact_1143_r_Ocarrier__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 ) ) ) ) ) ).
% r.carrier_one_zero
thf(fact_1144_r_Ocarrier__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 ) ) ) ) ) ).
% r.carrier_one_not_zero
thf(fact_1145_r_Oone__closed,axiom,
member_set_int @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% r.one_closed
thf(fact_1146_r_Or__one,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= X ) ) ).
% r.r_one
thf(fact_1147_r_Ol__one,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X )
= X ) ) ).
% r.l_one
thf(fact_1148_ring__iso__memE_I4_J,axiom,
! [H: set_int > set_int,R3: partia4934656038542163276t_unit,S2: partia4934656038542163276t_unit] :
( ( member5205197933313416826et_int @ H @ ( ring_i7857697894474155344t_unit @ R3 @ S2 ) )
=> ( ( H @ ( one_se8065767436706823081t_unit @ R3 ) )
= ( one_se8065767436706823081t_unit @ S2 ) ) ) ).
% ring_iso_memE(4)
thf(fact_1149_morphic__propE_I1_J,axiom,
! [R3: partia4934656038542163276t_unit,P: set_int > $o] :
( ( morphi7684586164509475305t_unit @ R3 @ P )
=> ( P @ ( one_se8065767436706823081t_unit @ R3 ) ) ) ).
% morphic_propE(1)
thf(fact_1150_morphic__propE_I1_J,axiom,
! [R3: partia4692342223508353374t_unit,P: nat > $o] :
( ( morphi2578836188448194427t_unit @ R3 @ P )
=> ( P @ ( one_na902338870878123981t_unit @ R3 ) ) ) ).
% morphic_propE(1)
thf(fact_1151_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_1152_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_1153_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_1154_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_1155_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_1156_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_1157_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_1158_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_1159_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_1160_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1161_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1162_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_1163_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_1164_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_1165_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_1166_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_1167_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1168_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_1169_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_1170_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1171_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_1172_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_1173_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N3: nat] :
( ( P4 @ N3 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ~ ( P4 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_1174_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_1175_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_1176_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_1177_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_1178_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_1179_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_1180_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_1181_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_1182_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_1183_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_1184_morphic__prop__def,axiom,
( morphi2578836188448194427t_unit
= ( ^ [R4: partia4692342223508353374t_unit,P4: nat > $o] :
( ( P4 @ ( one_na902338870878123981t_unit @ R4 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R4 ) )
=> ( P4 @ X2 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R4 ) )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R4 ) )
=> ( P4 @ ( mult_n6028127365542633569t_unit @ R4 @ X2 @ Y3 ) ) ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( partia3499330772048238685t_unit @ R4 ) )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( partia3499330772048238685t_unit @ R4 ) )
=> ( P4 @ ( add_nat_Product_unit @ R4 @ X2 @ Y3 ) ) ) ) ) ) ) ).
% morphic_prop_def
thf(fact_1185_morphic__prop__def,axiom,
( morphi7684586164509475305t_unit
= ( ^ [R4: partia4934656038542163276t_unit,P4: set_int > $o] :
( ( P4 @ ( one_se8065767436706823081t_unit @ R4 ) )
& ! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R4 ) )
=> ( P4 @ X2 ) )
& ! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R4 ) )
=> ! [Y3: set_int] :
( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R4 ) )
=> ( P4 @ ( mult_s3864001451298473021t_unit @ R4 @ X2 @ Y3 ) ) ) )
& ! [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R4 ) )
=> ! [Y3: set_int] :
( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ R4 ) )
=> ( P4 @ ( add_se5859248395121729892t_unit @ R4 @ X2 @ Y3 ) ) ) ) ) ) ) ).
% morphic_prop_def
thf(fact_1186_morphic__propI,axiom,
! [P: nat > $o,R3: partia4692342223508353374t_unit] :
( ( P @ ( one_na902338870878123981t_unit @ R3 ) )
=> ( ! [R5: nat] :
( ( member_nat @ R5 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( P @ R5 ) )
=> ( ! [R12: nat,R23: nat] :
( ( member_nat @ R12 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ R23 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( P @ ( mult_n6028127365542633569t_unit @ R3 @ R12 @ R23 ) ) ) )
=> ( ! [R12: nat,R23: nat] :
( ( member_nat @ R12 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( ( member_nat @ R23 @ ( partia3499330772048238685t_unit @ R3 ) )
=> ( P @ ( add_nat_Product_unit @ R3 @ R12 @ R23 ) ) ) )
=> ( morphi2578836188448194427t_unit @ R3 @ P ) ) ) ) ) ).
% morphic_propI
thf(fact_1187_morphic__propI,axiom,
! [P: set_int > $o,R3: partia4934656038542163276t_unit] :
( ( P @ ( one_se8065767436706823081t_unit @ R3 ) )
=> ( ! [R5: set_int] :
( ( member_set_int @ R5 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( P @ R5 ) )
=> ( ! [R12: set_int,R23: set_int] :
( ( member_set_int @ R12 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ R23 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( P @ ( mult_s3864001451298473021t_unit @ R3 @ R12 @ R23 ) ) ) )
=> ( ! [R12: set_int,R23: set_int] :
( ( member_set_int @ R12 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( ( member_set_int @ R23 @ ( partia966996272515721803t_unit @ R3 ) )
=> ( P @ ( add_se5859248395121729892t_unit @ R3 @ R12 @ R23 ) ) ) )
=> ( morphi7684586164509475305t_unit @ R3 @ P ) ) ) ) ) ).
% morphic_propI
thf(fact_1188_leD,axiom,
! [Y: set_set_int,X: set_set_int] :
( ( ord_le4403425263959731960et_int @ Y @ X )
=> ~ ( ord_less_set_set_int @ X @ Y ) ) ).
% leD
thf(fact_1189_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_1190_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_1191_nless__le,axiom,
! [A: set_set_int,B: set_set_int] :
( ( ~ ( ord_less_set_set_int @ A @ B ) )
= ( ~ ( ord_le4403425263959731960et_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1192_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1193_antisym__conv1,axiom,
! [X: set_set_int,Y: set_set_int] :
( ~ ( ord_less_set_set_int @ X @ Y )
=> ( ( ord_le4403425263959731960et_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1194_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1195_antisym__conv2,axiom,
! [X: set_set_int,Y: set_set_int] :
( ( ord_le4403425263959731960et_int @ X @ Y )
=> ( ( ~ ( ord_less_set_set_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1196_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1197_less__le__not__le,axiom,
( ord_less_set_set_int
= ( ^ [X2: set_set_int,Y3: set_set_int] :
( ( ord_le4403425263959731960et_int @ X2 @ Y3 )
& ~ ( ord_le4403425263959731960et_int @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1198_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1199_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1200_order_Oorder__iff__strict,axiom,
( ord_le4403425263959731960et_int
= ( ^ [A4: set_set_int,B3: set_set_int] :
( ( ord_less_set_set_int @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1201_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1202_order_Ostrict__iff__order,axiom,
( ord_less_set_set_int
= ( ^ [A4: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1203_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1204_order_Ostrict__trans1,axiom,
! [A: set_set_int,B: set_set_int,C: set_set_int] :
( ( ord_le4403425263959731960et_int @ A @ B )
=> ( ( ord_less_set_set_int @ B @ C )
=> ( ord_less_set_set_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1205_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1206_order_Ostrict__trans2,axiom,
! [A: set_set_int,B: set_set_int,C: set_set_int] :
( ( ord_less_set_set_int @ A @ B )
=> ( ( ord_le4403425263959731960et_int @ B @ C )
=> ( ord_less_set_set_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1207_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1208_order_Ostrict__iff__not,axiom,
( ord_less_set_set_int
= ( ^ [A4: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A4 @ B3 )
& ~ ( ord_le4403425263959731960et_int @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1209_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1210_dual__order_Oorder__iff__strict,axiom,
( ord_le4403425263959731960et_int
= ( ^ [B3: set_set_int,A4: set_set_int] :
( ( ord_less_set_set_int @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1211_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1212_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_set_int
= ( ^ [B3: set_set_int,A4: set_set_int] :
( ( ord_le4403425263959731960et_int @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1213_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1214_dual__order_Ostrict__trans1,axiom,
! [B: set_set_int,A: set_set_int,C: set_set_int] :
( ( ord_le4403425263959731960et_int @ B @ A )
=> ( ( ord_less_set_set_int @ C @ B )
=> ( ord_less_set_set_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1215_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1216_r_OsubdomainI,axiom,
! [H2: set_set_int] :
( ( subcri1024317279029940167t_unit @ H2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ! [H12: set_int,H23: set_int] :
( ( member_set_int @ H12 @ H2 )
=> ( ( member_set_int @ H23 @ H2 )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H12 @ H23 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( H12
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
| ( H23
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) )
=> ( subdom1520866149873910708t_unit @ H2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.subdomainI
thf(fact_1217_r_Osubfield__m__inv_I2_J,axiom,
! [K2: set_set_int,K: set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ K @ ( minus_8897228262479074673et_int @ K2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K ) )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.subfield_m_inv(2)
thf(fact_1218_r_Ocomm__inv__char,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X )
= Y ) ) ) ) ).
% r.comm_inv_char
thf(fact_1219_r_Oinv__char,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X )
= Y ) ) ) ) ) ).
% r.inv_char
thf(fact_1220_r_Oinv__unique_H,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( Y
= ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) ) ) ) ) ) ).
% r.inv_unique'
thf(fact_1221_r_Osubfield__m__inv_I1_J,axiom,
! [K2: set_set_int,K: set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ K @ ( minus_8897228262479074673et_int @ K2 @ ( 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 ) ) @ K ) @ ( minus_8897228262479074673et_int @ K2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) ) ) ) ).
% r.subfield_m_inv(1)
thf(fact_1222_r_Osubfield__m__inv_I3_J,axiom,
! [K2: set_set_int,K: set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ K @ ( minus_8897228262479074673et_int @ K2 @ ( 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 ) ) @ K ) @ K )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.subfield_m_inv(3)
thf(fact_1223_r_Oinv__one,axiom,
( ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.inv_one
thf(fact_1224_r_Ocring__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 ) ) ) )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A3 @ X4 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) )
=> ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.cring_fieldI2
thf(fact_1225_r_Oadd_Oone__in__subset,axiom,
! [H2: set_set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( H2 != bot_bot_set_set_int )
=> ( ! [X3: set_int] :
( ( member_set_int @ X3 @ H2 )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X3 ) @ H2 ) )
=> ( ! [X3: set_int] :
( ( member_set_int @ X3 @ H2 )
=> ! [Xa: set_int] :
( ( member_set_int @ Xa @ H2 )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X3 @ Xa ) @ H2 ) ) )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ H2 ) ) ) ) ) ).
% r.add.one_in_subset
thf(fact_1226_r_Osubring__props_I5_J,axiom,
! [K2: set_set_int,H: set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ H @ K2 )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H ) @ K2 ) ) ) ).
% r.subring_props(5)
thf(fact_1227_r_Oa__transpose__inv,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= Z )
=> ( ( member_set_int @ X @ ( 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 ) ) @ X ) @ Z )
= Y ) ) ) ) ) ).
% r.a_transpose_inv
thf(fact_1228_r_Oadd_Oinv__mult__group,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( 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 ) ) @ X @ 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 ) ) @ X ) ) ) ) ) ).
% r.add.inv_mult_group
thf(fact_1229_r_Oadd_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 ) ) ) ) ) ) ).
% r.add.inv_solve_left
thf(fact_1230_r_Oadd_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 ) ) ) ) ) ) ).
% r.add.inv_solve_left'
thf(fact_1231_r_Oadd_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 ) ) ) ) ) ) ).
% r.add.inv_solve_right
thf(fact_1232_r_Oadd_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 ) ) ) ) ) ) ).
% r.add.inv_solve_right'
thf(fact_1233_r_Ominus__add,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( 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 ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ) ) ).
% r.minus_add
thf(fact_1234_r_Or__neg1,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( 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 ) ) @ X ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= Y ) ) ) ).
% r.r_neg1
thf(fact_1235_r_Or__neg2,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ Y ) )
= Y ) ) ) ).
% r.r_neg2
thf(fact_1236_r_Ol__minus,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( 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 ) ) @ X ) @ Y )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) ) ) ) ) ).
% r.l_minus
thf(fact_1237_r_Or__minus,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) ) ) ) ) ).
% r.r_minus
thf(fact_1238_r_Oadd_Onat__pow__inv,axiom,
! [X: set_int,I: nat] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7583499734880473159it_nat @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X ) ) ) ) ).
% r.add.nat_pow_inv
thf(fact_1239_r_Oadd_Oint__pow__inv,axiom,
! [X: set_int,I: int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I @ X ) ) ) ) ).
% r.add.int_pow_inv
thf(fact_1240_r_Ospace__subgroup__props_I4_J,axiom,
! [K2: set_set_int,N: nat,E: set_set_int,V4: set_int] :
( ( subfie3888952257595785920t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( embedd646006463343340164t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ K2 @ E )
=> ( ( member_set_int @ V4 @ E )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ V4 ) @ E ) ) ) ) ).
% r.space_subgroup_props(4)
thf(fact_1241_r_Ol__neg,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.l_neg
thf(fact_1242_r_Ominus__equality,axiom,
! [Y: set_int,X: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X )
= Y ) ) ) ) ).
% r.minus_equality
thf(fact_1243_r_Or__neg,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.r_neg
thf(fact_1244_r_Osum__zero__eq__neg,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( X
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ) ) ).
% r.sum_zero_eq_neg
thf(fact_1245_r_Oadd_Oint__pow__diff,axiom,
! [X: set_int,N: int,M: int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( minus_minus_int @ N @ M ) @ X )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ N @ X ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_po7581009264371422883it_int @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ X ) ) ) ) ) ).
% r.add.int_pow_diff
thf(fact_1246_r_Oadd_Oinv__closed,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.add.inv_closed
thf(fact_1247_r_Ominus__minus,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) )
= X ) ) ).
% r.minus_minus
thf(fact_1248_r_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 ) ) ) ) ).
% r.minus_zero
thf(fact_1249_r_Oadd_Oinv__eq__1__iff,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( X
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.inv_eq_1_iff
thf(fact_1250_r_Oinv__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 ) ) ) ) ) ).
% r.inv_neg_one
thf(fact_1251_r_Ozeromaximalideal__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 ) ) ) ) ).
% r.zeromaximalideal_eq_field
thf(fact_1252_r_Ozeromaximalideal__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 ) ) ) ) ).
% r.zeromaximalideal_fieldI
thf(fact_1253_r_Ominus__eq,axiom,
! [X: set_int,Y: set_int] :
( ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ).
% r.minus_eq
thf(fact_1254_r_Ominus__closed,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( 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 ) ) @ X @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.minus_closed
thf(fact_1255_r_Or__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.r_right_minus_eq
thf(fact_1256_r_Omaximalideal__prime,axiom,
! [I4: set_set_int] :
( ( maxima6262477034536100350t_unit @ I4 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( primei350866878734230858t_unit @ I4 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.maximalideal_prime
thf(fact_1257_r_Ofield__intro2,axiom,
( ( ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ! [X3: set_int] :
( ( member_set_int @ X3 @ ( 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 @ X3 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
=> ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.field_intro2
thf(fact_1258_r_OUnits__closed,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.Units_closed
thf(fact_1259_r_Oinv__eq__imp__eq,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( 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 ) ) @ X )
= ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) )
=> ( X = Y ) ) ) ) ).
% r.inv_eq_imp_eq
thf(fact_1260_r_Ounit__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 ) ) ) ) ) ) ) ).
% r.unit_factor
thf(fact_1261_r_Oprod__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 ) ) ) ) ) ) ) ) ).
% r.prod_unit_r
thf(fact_1262_r_Oprod__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 ) ) ) ) ) ) ) ) ).
% r.prod_unit_l
thf(fact_1263_r_OUnits__inv__comm,axiom,
! [X: set_int,Y: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( 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 @ X )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.Units_inv_comm
thf(fact_1264_r_Oideal__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 ) ) ) ) ) ) ).
% r.ideal_eq_carrier_iff
thf(fact_1265_r_Oinv__eq__one__eq,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( X
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.inv_eq_one_eq
thf(fact_1266_r_OUnits__r__inv__ex,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [X3: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ X3 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.Units_r_inv_ex
thf(fact_1267_r_OUnits__l__inv__ex,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [X3: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X3 @ X )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.Units_l_inv_ex
thf(fact_1268_r_Oinv__eq__neg__one__eq,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( m_inv_4894562657074299959t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
= ( X
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.inv_eq_neg_one_eq
thf(fact_1269_r_Ocring__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 ) ) ) ) ).
% r.cring_fieldI
thf(fact_1270_r_OUnits__m__closed,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( 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 ) ) @ X @ Y ) @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.Units_m_closed
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_nat @ ( zfact_iso_inv @ n @ x ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ).
%------------------------------------------------------------------------------