TPTP Problem File: SLH0197^1.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_00061_001970__18230266_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1388 ( 626 unt; 116 typ; 0 def)
% Number of atoms : 3386 (1482 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 9895 ( 334 ~; 79 |; 200 &;8071 @)
% ( 0 <=>;1211 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 603 ( 603 >; 0 *; 0 +; 0 <<)
% Number of symbols : 109 ( 106 usr; 17 con; 0-3 aty)
% Number of variables : 3142 ( 141 ^;2891 !; 110 ?;3142 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:40:07.315
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
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__Int__Oint_Mt__Group__Omonoid__Omonoid____ext_It__Int__Oint_Mt__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J_J_J,type,
partia2818514838349642498t_unit: $tType ).
thf(ty_n_t__Group__Omonoid__Omonoid____ext_It__Int__Oint_Mt__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J_J,type,
monoid8431999971278595628t_unit: $tType ).
thf(ty_n_t__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J,type,
ring_e6626950497611839816t_unit: $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__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (106)
thf(sy_c_AbelCoset_OA__RCOSETS_001t__Int__Oint_001t__Product____Type__Ounit,type,
a_RCOS3445019769541752303t_unit: partia2818514838349642498t_unit > set_int > set_set_int ).
thf(sy_c_AbelCoset_Oa__r__coset_001t__Int__Oint_001t__Product____Type__Ounit,type,
a_r_co6205493800230438172t_unit: partia2818514838349642498t_unit > set_int > int > set_int ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Int__Oint_001t__Group__Omonoid__Omonoid____ext_It__Int__Oint_Mt__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J_J,type,
partia8426541738980984321t_unit: partia2818514838349642498t_unit > set_int ).
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,
partia966996272515721803t_unit: partia4934656038542163276t_unit > set_set_int ).
thf(sy_c_Congruence_Opartial__object_Opartial__object__ext_001t__Int__Oint_001t__Group__Omonoid__Omonoid____ext_It__Int__Oint_Mt__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J_J,type,
partia4118392927963588428t_unit: set_int > monoid8431999971278595628t_unit > partia2818514838349642498t_unit ).
thf(sy_c_Coset_Oorder_001t__Int__Oint_001t__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J,type,
order_3730749346064586230t_unit: partia2818514838349642498t_unit > nat ).
thf(sy_c_Factorial__Ring_Onormalization__semidom__class_Oprime_001t__Int__Oint,type,
factor1798656936486255268me_int: int > $o ).
thf(sy_c_Factorial__Ring_Onormalization__semidom__class_Oprime_001t__Nat__Onat,type,
factor1801147406995305544me_nat: nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
finite_finite_int: set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
finite6197958912794628473et_int: set_set_int > $o ).
thf(sy_c_Fun_Obij__betw_001t__Int__Oint_001t__Int__Oint,type,
bij_betw_int_int: ( int > int ) > set_int > set_int > $o ).
thf(sy_c_Fun_Obij__betw_001t__Int__Oint_001t__Nat__Onat,type,
bij_betw_int_nat: ( int > nat ) > set_int > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
bij_betw_int_set_int: ( int > set_int ) > set_int > set_set_int > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Int__Oint,type,
bij_betw_nat_int: ( nat > int ) > set_nat > set_int > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
bij_betw_nat_set_int: ( nat > set_int ) > set_nat > set_set_int > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Int__Oint_J_001t__Int__Oint,type,
bij_betw_set_int_int: ( set_int > int ) > set_set_int > set_int > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Int__Oint_J_001t__Nat__Onat,type,
bij_betw_set_int_nat: ( set_int > nat ) > set_set_int > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
bij_be5268973184346298300et_int: ( set_int > set_int ) > set_set_int > set_set_int > $o ).
thf(sy_c_Group_Omonoid_Omonoid__ext_001t__Int__Oint_001t__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J,type,
monoid6080699973261426200t_unit: ( int > int > int ) > int > ring_e6626950497611839816t_unit > monoid8431999971278595628t_unit ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > 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_Ideal_Ocgenideal_001t__Int__Oint_001t__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J,type,
cgenid7570434961818237563t_unit: partia2818514838349642498t_unit > int > set_int ).
thf(sy_c_Ideal_Ogenideal_001t__Int__Oint_001t__Product____Type__Ounit,type,
genide1613390280493775889t_unit: partia2818514838349642498t_unit > set_int > set_int ).
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_Oideal_001t__Int__Oint_001t__Product____Type__Ounit,type,
ideal_6787631597145370931t_unit: set_int > partia2818514838349642498t_unit > $o ).
thf(sy_c_Ideal_Omaximalideal_001t__Int__Oint_001t__Product____Type__Ounit,type,
maxima7040249999675607092t_unit: set_int > partia2818514838349642498t_unit > $o ).
thf(sy_c_Ideal_Oprimeideal_001t__Int__Oint_001t__Product____Type__Ounit,type,
primei2109666362732673920t_unit: set_int > partia2818514838349642498t_unit > $o ).
thf(sy_c_Ideal_Oprincipalideal_001t__Int__Oint_001t__Product____Type__Ounit,type,
princi1768892856804252751t_unit: set_int > partia2818514838349642498t_unit > $o ).
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_IntRing_OZMod,type,
zMod: int > int > set_int ).
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__Int__Oint_M_Eo_J,type,
bot_bot_int_o: int > $o ).
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_It__Int__Oint_J,type,
bot_bot_set_int: 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_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
ord_less_set_set_int: set_set_int > set_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Int__Oint_M_Eo_J,type,
top_top_int_o: int > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
top_top_set_int: set_int ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
top_top_set_set_int: set_set_int ).
thf(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
thf(sy_c_QuotRing_OFactRing_001t__Int__Oint_001t__Product____Type__Ounit,type,
factRi5755170488246124606t_unit: partia2818514838349642498t_unit > set_int > partia4934656038542163276t_unit ).
thf(sy_c_QuotRing_Ois__ring__iso_001t__Int__Oint_001t__Product____Type__Ounit_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
is_rin1886641436590440976t_unit: partia2818514838349642498t_unit > partia4934656038542163276t_unit > $o ).
thf(sy_c_QuotRing_Ois__ring__iso_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Int__Oint_001t__Product____Type__Ounit,type,
is_rin6476721666283997948t_unit: partia4934656038542163276t_unit > partia2818514838349642498t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__Int__Oint_001t__Product____Type__Ounit,type,
domain1430183510194609567t_unit: partia2818514838349642498t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
domain6183376680155302761t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__Int__Oint_001t__Product____Type__Ounit,type,
field_5117527561578272769t_unit: partia2818514838349642498t_unit > $o ).
thf(sy_c_Ring_Oring_Oring__ext_001t__Int__Oint_001t__Product____Type__Ounit,type,
ring_e5272872978682396362t_unit: int > ( int > int > int ) > product_unit > ring_e6626950497611839816t_unit ).
thf(sy_c_Ring__Characteristic_Ochar_001t__Int__Oint_001t__Product____Type__Ounit,type,
ring_c8451697193457390130t_unit: partia2818514838349642498t_unit > nat ).
thf(sy_c_Ring__Characteristic_Oint__embed_001t__Int__Oint_001t__Product____Type__Ounit,type,
ring_i2003929451149106142t_unit: partia2818514838349642498t_unit > int > int ).
thf(sy_c_Ring__Characteristic_Ozfact__iso,type,
ring_zfact_iso: nat > nat > set_int ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
modulo_modulo_int: int > int > int ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > 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_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
image_int_set_int: ( int > set_int ) > set_int > set_set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
image_nat_set_int: ( nat > set_int ) > set_nat > set_set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Int__Oint,type,
image_set_int_int: ( set_int > int ) > set_set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Nat__Onat,type,
image_set_int_nat: ( set_int > nat ) > set_set_int > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
image_524474410958335435et_int: ( set_int > set_int ) > set_set_int > set_set_int ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > 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_001t__Int__Oint,type,
set_ord_lessThan_int: int > 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_Osubdomain_001t__Int__Oint_001t__Product____Type__Ounit,type,
subdom8488461989912461802t_unit: set_int > partia2818514838349642498t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001t__Int__Oint_001t__Product____Type__Ounit,type,
subfie2008777110680908022t_unit: set_int > partia2818514838349642498t_unit > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: 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_I,type,
i: set_int ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_x,type,
x: set_int ).
% Relevant facts (1268)
thf(fact_0_n__ge__0,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% n_ge_0
thf(fact_1_assms_I2_J,axiom,
member_set_int @ x @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% assms(2)
thf(fact_2_zfact__iso__ran,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( image_nat_set_int @ ( ring_zfact_iso @ N ) @ ( set_ord_lessThan_nat @ N ) )
= ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% zfact_iso_ran
thf(fact_3_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_4_lessThan__eq__iff,axiom,
! [X: int,Y: int] :
( ( ( set_ord_lessThan_int @ X )
= ( set_ord_lessThan_int @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_5_image__eqI,axiom,
! [B: set_int,F: set_int > set_int,X: set_int,A: set_set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_set_int @ X @ A )
=> ( member_set_int @ B @ ( image_524474410958335435et_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_6_image__eqI,axiom,
! [B: int,F: set_int > int,X: set_int,A: set_set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_set_int @ X @ A )
=> ( member_int @ B @ ( image_set_int_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_7_image__eqI,axiom,
! [B: nat,F: set_int > nat,X: set_int,A: set_set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_set_int @ X @ A )
=> ( member_nat @ B @ ( image_set_int_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_8_image__eqI,axiom,
! [B: set_int,F: int > set_int,X: int,A: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A )
=> ( member_set_int @ B @ ( image_int_set_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_9_image__eqI,axiom,
! [B: int,F: int > int,X: int,A: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A )
=> ( member_int @ B @ ( image_int_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_10_image__eqI,axiom,
! [B: nat,F: int > nat,X: int,A: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A )
=> ( member_nat @ B @ ( image_int_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_11_image__eqI,axiom,
! [B: set_int,F: nat > set_int,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_set_int @ B @ ( image_nat_set_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_12_image__eqI,axiom,
! [B: int,F: nat > int,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_int @ B @ ( image_nat_int @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_13_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_14_Inf_OINF__cong,axiom,
! [A: set_set_int,B2: set_set_int,C: set_int > int,D: set_int > int,Inf: set_int > int] :
( ( A = B2 )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_set_int_int @ C @ A ) )
= ( Inf @ ( image_set_int_int @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_15_Inf_OINF__cong,axiom,
! [A: set_int,B2: set_int,C: int > set_int,D: int > set_int,Inf: set_set_int > set_int] :
( ( A = B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_int_set_int @ C @ A ) )
= ( Inf @ ( image_int_set_int @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_16_Inf_OINF__cong,axiom,
! [A: set_int,B2: set_int,C: int > nat,D: int > nat,Inf: set_nat > nat] :
( ( A = B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_int_nat @ C @ A ) )
= ( Inf @ ( image_int_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_17_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C @ A ) )
= ( Inf @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_18_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > int,D: nat > int,Inf: set_int > int] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_int @ C @ A ) )
= ( Inf @ ( image_nat_int @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_19_Inf_OINF__cong,axiom,
! [A: set_int,B2: set_int,C: int > int,D: int > int,Inf: set_int > int] :
( ( A = B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_int_int @ C @ A ) )
= ( Inf @ ( image_int_int @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_20_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > set_int,D: nat > set_int,Inf: set_set_int > set_int] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_set_int @ C @ A ) )
= ( Inf @ ( image_nat_set_int @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_21_Sup_OSUP__cong,axiom,
! [A: set_set_int,B2: set_set_int,C: set_int > int,D: set_int > int,Sup: set_int > int] :
( ( A = B2 )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_set_int_int @ C @ A ) )
= ( Sup @ ( image_set_int_int @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_22_Sup_OSUP__cong,axiom,
! [A: set_int,B2: set_int,C: int > set_int,D: int > set_int,Sup: set_set_int > set_int] :
( ( A = B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_int_set_int @ C @ A ) )
= ( Sup @ ( image_int_set_int @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_23_Sup_OSUP__cong,axiom,
! [A: set_int,B2: set_int,C: int > nat,D: int > nat,Sup: set_nat > nat] :
( ( A = B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_int_nat @ C @ A ) )
= ( Sup @ ( image_int_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_24_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C @ A ) )
= ( Sup @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_25_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > int,D: nat > int,Sup: set_int > int] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_int @ C @ A ) )
= ( Sup @ ( image_nat_int @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_26_Sup_OSUP__cong,axiom,
! [A: set_int,B2: set_int,C: int > int,D: int > int,Sup: set_int > int] :
( ( A = B2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_int_int @ C @ A ) )
= ( Sup @ ( image_int_int @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_27_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C: nat > set_int,D: nat > set_int,Sup: set_set_int > set_int] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_set_int @ C @ A ) )
= ( Sup @ ( image_nat_set_int @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_28_imageI,axiom,
! [X: set_int,A: set_set_int,F: set_int > set_int] :
( ( member_set_int @ X @ A )
=> ( member_set_int @ ( F @ X ) @ ( image_524474410958335435et_int @ F @ A ) ) ) ).
% imageI
thf(fact_29_imageI,axiom,
! [X: set_int,A: set_set_int,F: set_int > int] :
( ( member_set_int @ X @ A )
=> ( member_int @ ( F @ X ) @ ( image_set_int_int @ F @ A ) ) ) ).
% imageI
thf(fact_30_imageI,axiom,
! [X: set_int,A: set_set_int,F: set_int > nat] :
( ( member_set_int @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_set_int_nat @ F @ A ) ) ) ).
% imageI
thf(fact_31_imageI,axiom,
! [X: int,A: set_int,F: int > set_int] :
( ( member_int @ X @ A )
=> ( member_set_int @ ( F @ X ) @ ( image_int_set_int @ F @ A ) ) ) ).
% imageI
thf(fact_32_imageI,axiom,
! [X: int,A: set_int,F: int > int] :
( ( member_int @ X @ A )
=> ( member_int @ ( F @ X ) @ ( image_int_int @ F @ A ) ) ) ).
% imageI
thf(fact_33_imageI,axiom,
! [X: int,A: set_int,F: int > nat] :
( ( member_int @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_int_nat @ F @ A ) ) ) ).
% imageI
thf(fact_34_imageI,axiom,
! [X: nat,A: set_nat,F: nat > set_int] :
( ( member_nat @ X @ A )
=> ( member_set_int @ ( F @ X ) @ ( image_nat_set_int @ F @ A ) ) ) ).
% imageI
thf(fact_35_imageI,axiom,
! [X: nat,A: set_nat,F: nat > int] :
( ( member_nat @ X @ A )
=> ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ A ) ) ) ).
% imageI
thf(fact_36_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_37_image__iff,axiom,
! [Z: set_int,F: int > set_int,A: set_int] :
( ( member_set_int @ Z @ ( image_int_set_int @ F @ A ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_38_image__iff,axiom,
! [Z: int,F: set_int > int,A: set_set_int] :
( ( member_int @ Z @ ( image_set_int_int @ F @ A ) )
= ( ? [X3: set_int] :
( ( member_set_int @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_39_image__iff,axiom,
! [Z: int,F: nat > int,A: set_nat] :
( ( member_int @ Z @ ( image_nat_int @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_40_image__iff,axiom,
! [Z: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_41_image__iff,axiom,
! [Z: nat,F: int > nat,A: set_int] :
( ( member_nat @ Z @ ( image_int_nat @ F @ A ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_42_image__iff,axiom,
! [Z: set_int,F: nat > set_int,A: set_nat] :
( ( member_set_int @ Z @ ( image_nat_set_int @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_43_image__iff,axiom,
! [Z: int,F: int > int,A: set_int] :
( ( member_int @ Z @ ( image_int_int @ F @ A ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ A )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_44_bex__imageD,axiom,
! [F: set_int > int,A: set_set_int,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( image_set_int_int @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_45_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_46_bex__imageD,axiom,
! [F: nat > int,A: set_nat,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( image_nat_int @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_47_bex__imageD,axiom,
! [F: int > set_int,A: set_int,P: set_int > $o] :
( ? [X4: set_int] :
( ( member_set_int @ X4 @ ( image_int_set_int @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_48_bex__imageD,axiom,
! [F: int > nat,A: set_int,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_int_nat @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_49_bex__imageD,axiom,
! [F: nat > set_int,A: set_nat,P: set_int > $o] :
( ? [X4: set_int] :
( ( member_set_int @ X4 @ ( image_nat_set_int @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_50_bex__imageD,axiom,
! [F: int > int,A: set_int,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( image_int_int @ F @ A ) )
& ( P @ X4 ) )
=> ? [X2: int] :
( ( member_int @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_51_image__cong,axiom,
! [M: set_set_int,N2: set_set_int,F: set_int > int,G: set_int > int] :
( ( M = N2 )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_set_int_int @ F @ M )
= ( image_set_int_int @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_52_image__cong,axiom,
! [M: set_int,N2: set_int,F: int > int,G: int > int] :
( ( M = N2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_int_int @ F @ M )
= ( image_int_int @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_53_image__cong,axiom,
! [M: set_int,N2: set_int,F: int > set_int,G: int > set_int] :
( ( M = N2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_int_set_int @ F @ M )
= ( image_int_set_int @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_54_image__cong,axiom,
! [M: set_int,N2: set_int,F: int > nat,G: int > nat] :
( ( M = N2 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_int_nat @ F @ M )
= ( image_int_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_55_image__cong,axiom,
! [M: set_nat,N2: set_nat,F: nat > set_int,G: nat > set_int] :
( ( M = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_set_int @ F @ M )
= ( image_nat_set_int @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_56_image__cong,axiom,
! [M: set_nat,N2: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_57_image__cong,axiom,
! [M: set_nat,N2: set_nat,F: nat > int,G: nat > int] :
( ( M = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_int @ F @ M )
= ( image_nat_int @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_58_ball__imageD,axiom,
! [F: nat > set_int,A: set_nat,P: set_int > $o] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ ( image_nat_set_int @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_59_ball__imageD,axiom,
! [F: int > int,A: set_int,P: int > $o] :
( ! [X2: int] :
( ( member_int @ X2 @ ( image_int_int @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: int] :
( ( member_int @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_60_ball__imageD,axiom,
! [F: set_int > int,A: set_set_int,P: int > $o] :
( ! [X2: int] :
( ( member_int @ X2 @ ( image_set_int_int @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: set_int] :
( ( member_set_int @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_61_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_62_ball__imageD,axiom,
! [F: nat > int,A: set_nat,P: int > $o] :
( ! [X2: int] :
( ( member_int @ X2 @ ( image_nat_int @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_63_ball__imageD,axiom,
! [F: int > set_int,A: set_int,P: set_int > $o] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ ( image_int_set_int @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: int] :
( ( member_int @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_64_ball__imageD,axiom,
! [F: int > nat,A: set_int,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_int_nat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X4: int] :
( ( member_int @ X4 @ A )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_65_rev__image__eqI,axiom,
! [X: set_int,A: set_set_int,B: set_int,F: set_int > set_int] :
( ( member_set_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_set_int @ B @ ( image_524474410958335435et_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_66_rev__image__eqI,axiom,
! [X: set_int,A: set_set_int,B: int,F: set_int > int] :
( ( member_set_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_set_int_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_67_rev__image__eqI,axiom,
! [X: set_int,A: set_set_int,B: nat,F: set_int > nat] :
( ( member_set_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_set_int_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_68_rev__image__eqI,axiom,
! [X: int,A: set_int,B: set_int,F: int > set_int] :
( ( member_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_set_int @ B @ ( image_int_set_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_69_rev__image__eqI,axiom,
! [X: int,A: set_int,B: int,F: int > int] :
( ( member_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_int_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_70_rev__image__eqI,axiom,
! [X: int,A: set_int,B: nat,F: int > nat] :
( ( member_int @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_int_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_71_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: set_int,F: nat > set_int] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_set_int @ B @ ( image_nat_set_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_72_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: int,F: nat > int] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_nat_int @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_73_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_74_image__set__eqI,axiom,
! [A: set_set_int,F: set_int > set_int,B2: set_set_int,G: set_int > set_int] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ A )
=> ( member_set_int @ ( F @ X2 ) @ B2 ) )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ B2 )
=> ( ( member_set_int @ ( G @ X2 ) @ A )
& ( ( F @ ( G @ X2 ) )
= X2 ) ) )
=> ( ( image_524474410958335435et_int @ F @ A )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_75_image__set__eqI,axiom,
! [A: set_set_int,F: set_int > int,B2: set_int,G: int > set_int] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ A )
=> ( member_int @ ( F @ X2 ) @ B2 ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B2 )
=> ( ( member_set_int @ ( G @ X2 ) @ A )
& ( ( F @ ( G @ X2 ) )
= X2 ) ) )
=> ( ( image_set_int_int @ F @ A )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_76_image__set__eqI,axiom,
! [A: set_set_int,F: set_int > nat,B2: set_nat,G: nat > set_int] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( member_set_int @ ( G @ X2 ) @ A )
& ( ( F @ ( G @ X2 ) )
= X2 ) ) )
=> ( ( image_set_int_nat @ F @ A )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_77_image__set__eqI,axiom,
! [A: set_int,F: int > set_int,B2: set_set_int,G: set_int > int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_set_int @ ( F @ X2 ) @ B2 ) )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ B2 )
=> ( ( member_int @ ( G @ X2 ) @ A )
& ( ( F @ ( G @ X2 ) )
= X2 ) ) )
=> ( ( image_int_set_int @ F @ A )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_78_image__set__eqI,axiom,
! [A: set_int,F: int > int,B2: set_int,G: int > int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_int @ ( F @ X2 ) @ B2 ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B2 )
=> ( ( member_int @ ( G @ X2 ) @ A )
& ( ( F @ ( G @ X2 ) )
= X2 ) ) )
=> ( ( image_int_int @ F @ A )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_79_image__set__eqI,axiom,
! [A: set_int,F: int > nat,B2: set_nat,G: nat > int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( member_int @ ( G @ X2 ) @ A )
& ( ( F @ ( G @ X2 ) )
= X2 ) ) )
=> ( ( image_int_nat @ F @ A )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_80_image__set__eqI,axiom,
! [A: set_nat,F: nat > set_int,B2: set_set_int,G: set_int > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_set_int @ ( F @ X2 ) @ B2 ) )
=> ( ! [X2: set_int] :
( ( member_set_int @ X2 @ B2 )
=> ( ( member_nat @ ( G @ X2 ) @ A )
& ( ( F @ ( G @ X2 ) )
= X2 ) ) )
=> ( ( image_nat_set_int @ F @ A )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_81_image__set__eqI,axiom,
! [A: set_nat,F: nat > int,B2: set_int,G: int > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_int @ ( F @ X2 ) @ B2 ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ B2 )
=> ( ( member_nat @ ( G @ X2 ) @ A )
& ( ( F @ ( G @ X2 ) )
= X2 ) ) )
=> ( ( image_nat_int @ F @ A )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_82_image__set__eqI,axiom,
! [A: set_nat,F: nat > nat,B2: set_nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( member_nat @ ( G @ X2 ) @ A )
& ( ( F @ ( G @ X2 ) )
= X2 ) ) )
=> ( ( image_nat_nat @ F @ A )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_83_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_84_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_85_lessThan__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I @ K ) ) ).
% lessThan_iff
thf(fact_86_lessThan__strict__subset__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_87_lessThan__strict__subset__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M2 ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M2 @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_88_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_89_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_90_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_91_image__add__0,axiom,
! [S: set_int] :
( ( image_int_int @ ( plus_plus_int @ zero_zero_int ) @ S )
= S ) ).
% image_add_0
thf(fact_92_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_93_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_94_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_95_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_96_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_97_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_98_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_99_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_100_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_101_image__is__empty,axiom,
! [F: int > nat,A: set_int] :
( ( ( image_int_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_int ) ) ).
% image_is_empty
thf(fact_102_image__is__empty,axiom,
! [F: set_int > nat,A: set_set_int] :
( ( ( image_set_int_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_set_int ) ) ).
% image_is_empty
thf(fact_103_image__is__empty,axiom,
! [F: nat > int,A: set_nat] :
( ( ( image_nat_int @ F @ A )
= bot_bot_set_int )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_104_image__is__empty,axiom,
! [F: int > int,A: set_int] :
( ( ( image_int_int @ F @ A )
= bot_bot_set_int )
= ( A = bot_bot_set_int ) ) ).
% image_is_empty
thf(fact_105_image__is__empty,axiom,
! [F: set_int > int,A: set_set_int] :
( ( ( image_set_int_int @ F @ A )
= bot_bot_set_int )
= ( A = bot_bot_set_set_int ) ) ).
% image_is_empty
thf(fact_106_image__is__empty,axiom,
! [F: nat > set_int,A: set_nat] :
( ( ( image_nat_set_int @ F @ A )
= bot_bot_set_set_int )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_107_image__is__empty,axiom,
! [F: int > set_int,A: set_int] :
( ( ( image_int_set_int @ F @ A )
= bot_bot_set_set_int )
= ( A = bot_bot_set_int ) ) ).
% image_is_empty
thf(fact_108_image__is__empty,axiom,
! [F: set_int > set_int,A: set_set_int] :
( ( ( image_524474410958335435et_int @ F @ A )
= bot_bot_set_set_int )
= ( A = bot_bot_set_set_int ) ) ).
% image_is_empty
thf(fact_109_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_110_empty__is__image,axiom,
! [F: int > nat,A: set_int] :
( ( bot_bot_set_nat
= ( image_int_nat @ F @ A ) )
= ( A = bot_bot_set_int ) ) ).
% empty_is_image
thf(fact_111_empty__is__image,axiom,
! [F: set_int > nat,A: set_set_int] :
( ( bot_bot_set_nat
= ( image_set_int_nat @ F @ A ) )
= ( A = bot_bot_set_set_int ) ) ).
% empty_is_image
thf(fact_112_empty__is__image,axiom,
! [F: nat > int,A: set_nat] :
( ( bot_bot_set_int
= ( image_nat_int @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_113_empty__is__image,axiom,
! [F: int > int,A: set_int] :
( ( bot_bot_set_int
= ( image_int_int @ F @ A ) )
= ( A = bot_bot_set_int ) ) ).
% empty_is_image
thf(fact_114_empty__is__image,axiom,
! [F: set_int > int,A: set_set_int] :
( ( bot_bot_set_int
= ( image_set_int_int @ F @ A ) )
= ( A = bot_bot_set_set_int ) ) ).
% empty_is_image
thf(fact_115_empty__is__image,axiom,
! [F: nat > set_int,A: set_nat] :
( ( bot_bot_set_set_int
= ( image_nat_set_int @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_116_empty__is__image,axiom,
! [F: int > set_int,A: set_int] :
( ( bot_bot_set_set_int
= ( image_int_set_int @ F @ A ) )
= ( A = bot_bot_set_int ) ) ).
% empty_is_image
thf(fact_117_empty__is__image,axiom,
! [F: set_int > set_int,A: set_set_int] :
( ( bot_bot_set_set_int
= ( image_524474410958335435et_int @ F @ A ) )
= ( A = bot_bot_set_set_int ) ) ).
% empty_is_image
thf(fact_118_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_119_image__empty,axiom,
! [F: nat > int] :
( ( image_nat_int @ F @ bot_bot_set_nat )
= bot_bot_set_int ) ).
% image_empty
thf(fact_120_image__empty,axiom,
! [F: nat > set_int] :
( ( image_nat_set_int @ F @ bot_bot_set_nat )
= bot_bot_set_set_int ) ).
% image_empty
thf(fact_121_image__empty,axiom,
! [F: int > nat] :
( ( image_int_nat @ F @ bot_bot_set_int )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_122_image__empty,axiom,
! [F: int > int] :
( ( image_int_int @ F @ bot_bot_set_int )
= bot_bot_set_int ) ).
% image_empty
thf(fact_123_image__empty,axiom,
! [F: int > set_int] :
( ( image_int_set_int @ F @ bot_bot_set_int )
= bot_bot_set_set_int ) ).
% image_empty
thf(fact_124_image__empty,axiom,
! [F: set_int > nat] :
( ( image_set_int_nat @ F @ bot_bot_set_set_int )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_125_image__empty,axiom,
! [F: set_int > int] :
( ( image_set_int_int @ F @ bot_bot_set_set_int )
= bot_bot_set_int ) ).
% image_empty
thf(fact_126_image__empty,axiom,
! [F: set_int > set_int] :
( ( image_524474410958335435et_int @ F @ bot_bot_set_set_int )
= bot_bot_set_set_int ) ).
% image_empty
thf(fact_127_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_128_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_129_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_130_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_131_empty__Collect__eq,axiom,
! [P: int > $o] :
( ( bot_bot_set_int
= ( collect_int @ P ) )
= ( ! [X3: int] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_132_empty__Collect__eq,axiom,
! [P: set_int > $o] :
( ( bot_bot_set_set_int
= ( collect_set_int @ P ) )
= ( ! [X3: set_int] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_133_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_134_Collect__empty__eq,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( ! [X3: int] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_135_Collect__empty__eq,axiom,
! [P: set_int > $o] :
( ( ( collect_set_int @ P )
= bot_bot_set_set_int )
= ( ! [X3: set_int] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_136_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_137_all__not__in__conv,axiom,
! [A: set_int] :
( ( ! [X3: int] :
~ ( member_int @ X3 @ A ) )
= ( A = bot_bot_set_int ) ) ).
% all_not_in_conv
thf(fact_138_all__not__in__conv,axiom,
! [A: set_set_int] :
( ( ! [X3: set_int] :
~ ( member_set_int @ X3 @ A ) )
= ( A = bot_bot_set_set_int ) ) ).
% all_not_in_conv
thf(fact_139_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_140_empty__iff,axiom,
! [C2: int] :
~ ( member_int @ C2 @ bot_bot_set_int ) ).
% empty_iff
thf(fact_141_empty__iff,axiom,
! [C2: set_int] :
~ ( member_set_int @ C2 @ bot_bot_set_set_int ) ).
% empty_iff
thf(fact_142_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_143_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_144_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_145_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_146_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_147_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_148_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_149_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_150_psubsetD,axiom,
! [A: set_set_int,B2: set_set_int,C2: set_int] :
( ( ord_less_set_set_int @ A @ B2 )
=> ( ( member_set_int @ C2 @ A )
=> ( member_set_int @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_151_psubsetD,axiom,
! [A: set_int,B2: set_int,C2: int] :
( ( ord_less_set_int @ A @ B2 )
=> ( ( member_int @ C2 @ A )
=> ( member_int @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_152_psubsetD,axiom,
! [A: set_nat,B2: set_nat,C2: nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B2 ) ) ) ).
% psubsetD
thf(fact_153_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_154_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_155_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_156_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_157_mem__Collect__eq,axiom,
! [A2: set_int,P: set_int > $o] :
( ( member_set_int @ A2 @ ( collect_set_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_158_mem__Collect__eq,axiom,
! [A2: int,P: int > $o] :
( ( member_int @ A2 @ ( collect_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_159_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_160_Collect__mem__eq,axiom,
! [A: set_set_int] :
( ( collect_set_int
@ ^ [X3: set_int] : ( member_set_int @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_161_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_162_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_163_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_164_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_165_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_166_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_167_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_168_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_169_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_170_not__psubset__empty,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_171_not__psubset__empty,axiom,
! [A: set_int] :
~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% not_psubset_empty
thf(fact_172_not__psubset__empty,axiom,
! [A: set_set_int] :
~ ( ord_less_set_set_int @ A @ bot_bot_set_set_int ) ).
% not_psubset_empty
thf(fact_173_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_174_ex__in__conv,axiom,
! [A: set_int] :
( ( ? [X3: int] : ( member_int @ X3 @ A ) )
= ( A != bot_bot_set_int ) ) ).
% ex_in_conv
thf(fact_175_ex__in__conv,axiom,
! [A: set_set_int] :
( ( ? [X3: set_int] : ( member_set_int @ X3 @ A ) )
= ( A != bot_bot_set_set_int ) ) ).
% ex_in_conv
thf(fact_176_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_177_equals0I,axiom,
! [A: set_int] :
( ! [Y2: int] :
~ ( member_int @ Y2 @ A )
=> ( A = bot_bot_set_int ) ) ).
% equals0I
thf(fact_178_equals0I,axiom,
! [A: set_set_int] :
( ! [Y2: set_int] :
~ ( member_set_int @ Y2 @ A )
=> ( A = bot_bot_set_set_int ) ) ).
% equals0I
thf(fact_179_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_180_equals0D,axiom,
! [A: set_int,A2: int] :
( ( A = bot_bot_set_int )
=> ~ ( member_int @ A2 @ A ) ) ).
% equals0D
thf(fact_181_equals0D,axiom,
! [A: set_set_int,A2: set_int] :
( ( A = bot_bot_set_set_int )
=> ~ ( member_set_int @ A2 @ A ) ) ).
% equals0D
thf(fact_182_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_183_emptyE,axiom,
! [A2: int] :
~ ( member_int @ A2 @ bot_bot_set_int ) ).
% emptyE
thf(fact_184_emptyE,axiom,
! [A2: set_int] :
~ ( member_set_int @ A2 @ bot_bot_set_set_int ) ).
% emptyE
thf(fact_185_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_186_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_187_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_188_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_189_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_190_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_191_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_192_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_193_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_194_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_195_lessThan__non__empty,axiom,
! [X: int] :
( ( set_ord_lessThan_int @ X )
!= bot_bot_set_int ) ).
% lessThan_non_empty
thf(fact_196_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_197_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_198_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_199_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_200_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_201_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_202_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_203_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_204_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_205_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_206_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_207_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_208_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_209_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_210_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_211_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_212_less__add__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_213_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_214_less__add__same__cancel1,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_215_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_216_add__less__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_217_add__less__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_218_add__less__same__cancel1,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_219_add__less__cancel__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_220_add__less__cancel__right,axiom,
! [A2: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_221_add__less__cancel__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_222_add__less__cancel__left,axiom,
! [C2: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_int @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_223_double__eq__0__iff,axiom,
! [A2: int] :
( ( ( plus_plus_int @ A2 @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_224_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_225_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_226_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_227_add__left__cancel,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_228_add__left__cancel,axiom,
! [A2: int,B: int,C2: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_229_add__right__cancel,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C2 @ A2 ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_230_add__right__cancel,axiom,
! [B: int,A2: int,C2: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C2 @ A2 ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_231_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_232_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_233_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_234_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_235_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_236_add__cancel__left__left,axiom,
! [B: int,A2: int] :
( ( ( plus_plus_int @ B @ A2 )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_237_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_238_add__cancel__left__right,axiom,
! [A2: int,B: int] :
( ( ( plus_plus_int @ A2 @ B )
= A2 )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_239_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_240_add__cancel__right__left,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ B @ A2 ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_241_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_242_add__cancel__right__right,axiom,
! [A2: int,B: int] :
( ( A2
= ( plus_plus_int @ A2 @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_243_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_244_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_245_bot__set__def,axiom,
( bot_bot_set_int
= ( collect_int @ bot_bot_int_o ) ) ).
% bot_set_def
thf(fact_246_bot__set__def,axiom,
( bot_bot_set_set_int
= ( collect_set_int @ bot_bot_set_int_o ) ) ).
% bot_set_def
thf(fact_247_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_248_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_249_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_250_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_251_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C2 )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_252_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_253_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_254_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_255_group__cancel_Oadd1,axiom,
! [A: int,K: int,A2: int,B: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_256_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_257_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A2: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_258_add_Oassoc,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_259_add_Oassoc,axiom,
! [A2: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B ) @ C2 )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_260_add_Oleft__cancel,axiom,
! [A2: int,B: int,C2: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_261_add_Oright__cancel,axiom,
! [B: int,A2: int,C2: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C2 @ A2 ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_262_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_263_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_264_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_265_add_Oleft__commute,axiom,
! [B: int,A2: int,C2: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A2 @ C2 ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_266_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_267_add__left__imp__eq,axiom,
! [A2: int,B: int,C2: int] :
( ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ A2 @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_268_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C2 @ A2 ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_269_add__right__imp__eq,axiom,
! [B: int,A2: int,C2: int] :
( ( ( plus_plus_int @ B @ A2 )
= ( plus_plus_int @ C2 @ A2 ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_270_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_271_zadd__int__left,axiom,
! [M2: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_272_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_273_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_274_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_275_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_276_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_277_comm__monoid__add__class_Oadd__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_278_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_279_add_Ocomm__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.comm_neutral
thf(fact_280_add_Ogroup__left__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_281_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_282_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_283_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_284_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_285_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_286_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_287_add__strict__mono,axiom,
! [A2: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_288_add__strict__mono,axiom,
! [A2: int,B: int,C2: int,D2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ C2 @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_289_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_290_add__strict__left__mono,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_291_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_292_add__strict__right__mono,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_293_add__less__imp__less__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_294_add__less__imp__less__left,axiom,
! [C2: int,A2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_295_add__less__imp__less__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_296_add__less__imp__less__right,axiom,
! [A2: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_int @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_297_add__neg__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_298_add__neg__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_299_add__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_300_add__pos__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_301_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A2 @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_302_pos__add__strict,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_303_pos__add__strict,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A2 @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_304_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_305_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_306_insert__image,axiom,
! [X: set_int,A: set_set_int,F: set_int > int] :
( ( member_set_int @ X @ A )
=> ( ( insert_int @ ( F @ X ) @ ( image_set_int_int @ F @ A ) )
= ( image_set_int_int @ F @ A ) ) ) ).
% insert_image
thf(fact_307_insert__image,axiom,
! [X: set_int,A: set_set_int,F: set_int > set_int] :
( ( member_set_int @ X @ A )
=> ( ( insert_set_int @ ( F @ X ) @ ( image_524474410958335435et_int @ F @ A ) )
= ( image_524474410958335435et_int @ F @ A ) ) ) ).
% insert_image
thf(fact_308_insert__image,axiom,
! [X: int,A: set_int,F: int > nat] :
( ( member_int @ X @ A )
=> ( ( insert_nat @ ( F @ X ) @ ( image_int_nat @ F @ A ) )
= ( image_int_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_309_insert__image,axiom,
! [X: int,A: set_int,F: int > int] :
( ( member_int @ X @ A )
=> ( ( insert_int @ ( F @ X ) @ ( image_int_int @ F @ A ) )
= ( image_int_int @ F @ A ) ) ) ).
% insert_image
thf(fact_310_insert__image,axiom,
! [X: int,A: set_int,F: int > set_int] :
( ( member_int @ X @ A )
=> ( ( insert_set_int @ ( F @ X ) @ ( image_int_set_int @ F @ A ) )
= ( image_int_set_int @ F @ A ) ) ) ).
% insert_image
thf(fact_311_insert__image,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( insert_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) )
= ( image_nat_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_312_insert__image,axiom,
! [X: nat,A: set_nat,F: nat > int] :
( ( member_nat @ X @ A )
=> ( ( insert_int @ ( F @ X ) @ ( image_nat_int @ F @ A ) )
= ( image_nat_int @ F @ A ) ) ) ).
% insert_image
thf(fact_313_insert__image,axiom,
! [X: nat,A: set_nat,F: nat > set_int] :
( ( member_nat @ X @ A )
=> ( ( insert_set_int @ ( F @ X ) @ ( image_nat_set_int @ F @ A ) )
= ( image_nat_set_int @ F @ A ) ) ) ).
% insert_image
thf(fact_314_image__insert,axiom,
! [F: nat > nat,A2: nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A2 @ B2 ) )
= ( insert_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_315_image__insert,axiom,
! [F: nat > int,A2: nat,B2: set_nat] :
( ( image_nat_int @ F @ ( insert_nat @ A2 @ B2 ) )
= ( insert_int @ ( F @ A2 ) @ ( image_nat_int @ F @ B2 ) ) ) ).
% image_insert
thf(fact_316_image__insert,axiom,
! [F: nat > set_int,A2: nat,B2: set_nat] :
( ( image_nat_set_int @ F @ ( insert_nat @ A2 @ B2 ) )
= ( insert_set_int @ ( F @ A2 ) @ ( image_nat_set_int @ F @ B2 ) ) ) ).
% image_insert
thf(fact_317_image__insert,axiom,
! [F: int > nat,A2: int,B2: set_int] :
( ( image_int_nat @ F @ ( insert_int @ A2 @ B2 ) )
= ( insert_nat @ ( F @ A2 ) @ ( image_int_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_318_image__insert,axiom,
! [F: int > int,A2: int,B2: set_int] :
( ( image_int_int @ F @ ( insert_int @ A2 @ B2 ) )
= ( insert_int @ ( F @ A2 ) @ ( image_int_int @ F @ B2 ) ) ) ).
% image_insert
thf(fact_319_image__insert,axiom,
! [F: int > set_int,A2: int,B2: set_int] :
( ( image_int_set_int @ F @ ( insert_int @ A2 @ B2 ) )
= ( insert_set_int @ ( F @ A2 ) @ ( image_int_set_int @ F @ B2 ) ) ) ).
% image_insert
thf(fact_320_image__insert,axiom,
! [F: set_int > int,A2: set_int,B2: set_set_int] :
( ( image_set_int_int @ F @ ( insert_set_int @ A2 @ B2 ) )
= ( insert_int @ ( F @ A2 ) @ ( image_set_int_int @ F @ B2 ) ) ) ).
% image_insert
thf(fact_321_image__insert,axiom,
! [F: set_int > set_int,A2: set_int,B2: set_set_int] :
( ( image_524474410958335435et_int @ F @ ( insert_set_int @ A2 @ B2 ) )
= ( insert_set_int @ ( F @ A2 ) @ ( image_524474410958335435et_int @ F @ B2 ) ) ) ).
% image_insert
thf(fact_322_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_323_singletonI,axiom,
! [A2: int] : ( member_int @ A2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) ).
% singletonI
thf(fact_324_singletonI,axiom,
! [A2: set_int] : ( member_set_int @ A2 @ ( insert_set_int @ A2 @ bot_bot_set_set_int ) ) ).
% singletonI
thf(fact_325_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_326_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_327_zfact__iso__bij,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( bij_betw_nat_set_int @ ( ring_zfact_iso @ N ) @ ( set_ord_lessThan_nat @ N ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% zfact_iso_bij
thf(fact_328_int__ops_I5_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_329_int__plus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_330_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_331_insert__absorb2,axiom,
! [X: int,A: set_int] :
( ( insert_int @ X @ ( insert_int @ X @ A ) )
= ( insert_int @ X @ A ) ) ).
% insert_absorb2
thf(fact_332_insert__absorb2,axiom,
! [X: set_int,A: set_set_int] :
( ( insert_set_int @ X @ ( insert_set_int @ X @ A ) )
= ( insert_set_int @ X @ A ) ) ).
% insert_absorb2
thf(fact_333_insert__iff,axiom,
! [A2: set_int,B: set_int,A: set_set_int] :
( ( member_set_int @ A2 @ ( insert_set_int @ B @ A ) )
= ( ( A2 = B )
| ( member_set_int @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_334_insert__iff,axiom,
! [A2: int,B: int,A: set_int] :
( ( member_int @ A2 @ ( insert_int @ B @ A ) )
= ( ( A2 = B )
| ( member_int @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_335_insert__iff,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_336_insertCI,axiom,
! [A2: set_int,B2: set_set_int,B: set_int] :
( ( ~ ( member_set_int @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_set_int @ A2 @ ( insert_set_int @ B @ B2 ) ) ) ).
% insertCI
thf(fact_337_insertCI,axiom,
! [A2: int,B2: set_int,B: int] :
( ( ~ ( member_int @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_int @ A2 @ ( insert_int @ B @ B2 ) ) ) ).
% insertCI
thf(fact_338_insertCI,axiom,
! [A2: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_339_mult__cancel__right,axiom,
! [A2: int,C2: int,B: int] :
( ( ( times_times_int @ A2 @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_340_mult__cancel__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_341_mult__cancel__left,axiom,
! [C2: int,A2: int,B: int] :
( ( ( times_times_int @ C2 @ A2 )
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_342_mult__cancel__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ( times_times_nat @ C2 @ A2 )
= ( times_times_nat @ C2 @ B ) )
= ( ( C2 = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_343_mult__eq__0__iff,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_344_mult__eq__0__iff,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_345_mult__zero__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_346_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_347_mult__zero__left,axiom,
! [A2: int] :
( ( times_times_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_348_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_349_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_350_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_351_bij__betw__of__nat,axiom,
! [N2: set_nat,A: set_nat] :
( ( bij_betw_nat_nat @ semiri1316708129612266289at_nat @ N2 @ A )
= ( ( image_nat_nat @ semiri1316708129612266289at_nat @ N2 )
= A ) ) ).
% bij_betw_of_nat
thf(fact_352_bij__betw__of__nat,axiom,
! [N2: set_nat,A: set_int] :
( ( bij_betw_nat_int @ semiri1314217659103216013at_int @ N2 @ A )
= ( ( image_nat_int @ semiri1314217659103216013at_int @ N2 )
= A ) ) ).
% bij_betw_of_nat
thf(fact_353_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_354_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_355_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_356_mk__disjoint__insert,axiom,
! [A2: set_int,A: set_set_int] :
( ( member_set_int @ A2 @ A )
=> ? [B4: set_set_int] :
( ( A
= ( insert_set_int @ A2 @ B4 ) )
& ~ ( member_set_int @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_357_mk__disjoint__insert,axiom,
! [A2: int,A: set_int] :
( ( member_int @ A2 @ A )
=> ? [B4: set_int] :
( ( A
= ( insert_int @ A2 @ B4 ) )
& ~ ( member_int @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_358_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B4: set_nat] :
( ( A
= ( insert_nat @ A2 @ B4 ) )
& ~ ( member_nat @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_359_insert__commute,axiom,
! [X: int,Y: int,A: set_int] :
( ( insert_int @ X @ ( insert_int @ Y @ A ) )
= ( insert_int @ Y @ ( insert_int @ X @ A ) ) ) ).
% insert_commute
thf(fact_360_insert__commute,axiom,
! [X: set_int,Y: set_int,A: set_set_int] :
( ( insert_set_int @ X @ ( insert_set_int @ Y @ A ) )
= ( insert_set_int @ Y @ ( insert_set_int @ X @ A ) ) ) ).
% insert_commute
thf(fact_361_insert__eq__iff,axiom,
! [A2: set_int,A: set_set_int,B: set_int,B2: set_set_int] :
( ~ ( member_set_int @ A2 @ A )
=> ( ~ ( member_set_int @ B @ B2 )
=> ( ( ( insert_set_int @ A2 @ A )
= ( insert_set_int @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C4: set_set_int] :
( ( A
= ( insert_set_int @ B @ C4 ) )
& ~ ( member_set_int @ B @ C4 )
& ( B2
= ( insert_set_int @ A2 @ C4 ) )
& ~ ( member_set_int @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_362_insert__eq__iff,axiom,
! [A2: int,A: set_int,B: int,B2: set_int] :
( ~ ( member_int @ A2 @ A )
=> ( ~ ( member_int @ B @ B2 )
=> ( ( ( insert_int @ A2 @ A )
= ( insert_int @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C4: set_int] :
( ( A
= ( insert_int @ B @ C4 ) )
& ~ ( member_int @ B @ C4 )
& ( B2
= ( insert_int @ A2 @ C4 ) )
& ~ ( member_int @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_363_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C4: set_nat] :
( ( A
= ( insert_nat @ B @ C4 ) )
& ~ ( member_nat @ B @ C4 )
& ( B2
= ( insert_nat @ A2 @ C4 ) )
& ~ ( member_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_364_insert__absorb,axiom,
! [A2: set_int,A: set_set_int] :
( ( member_set_int @ A2 @ A )
=> ( ( insert_set_int @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_365_insert__absorb,axiom,
! [A2: int,A: set_int] :
( ( member_int @ A2 @ A )
=> ( ( insert_int @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_366_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_367_insert__ident,axiom,
! [X: set_int,A: set_set_int,B2: set_set_int] :
( ~ ( member_set_int @ X @ A )
=> ( ~ ( member_set_int @ X @ B2 )
=> ( ( ( insert_set_int @ X @ A )
= ( insert_set_int @ X @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_368_insert__ident,axiom,
! [X: int,A: set_int,B2: set_int] :
( ~ ( member_int @ X @ A )
=> ( ~ ( member_int @ X @ B2 )
=> ( ( ( insert_int @ X @ A )
= ( insert_int @ X @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_369_insert__ident,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat @ X @ A )
= ( insert_nat @ X @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_370_Set_Oset__insert,axiom,
! [X: set_int,A: set_set_int] :
( ( member_set_int @ X @ A )
=> ~ ! [B4: set_set_int] :
( ( A
= ( insert_set_int @ X @ B4 ) )
=> ( member_set_int @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_371_Set_Oset__insert,axiom,
! [X: int,A: set_int] :
( ( member_int @ X @ A )
=> ~ ! [B4: set_int] :
( ( A
= ( insert_int @ X @ B4 ) )
=> ( member_int @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_372_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ~ ! [B4: set_nat] :
( ( A
= ( insert_nat @ X @ B4 ) )
=> ( member_nat @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_373_insertI2,axiom,
! [A2: set_int,B2: set_set_int,B: set_int] :
( ( member_set_int @ A2 @ B2 )
=> ( member_set_int @ A2 @ ( insert_set_int @ B @ B2 ) ) ) ).
% insertI2
thf(fact_374_insertI2,axiom,
! [A2: int,B2: set_int,B: int] :
( ( member_int @ A2 @ B2 )
=> ( member_int @ A2 @ ( insert_int @ B @ B2 ) ) ) ).
% insertI2
thf(fact_375_insertI2,axiom,
! [A2: nat,B2: set_nat,B: nat] :
( ( member_nat @ A2 @ B2 )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_376_insertI1,axiom,
! [A2: set_int,B2: set_set_int] : ( member_set_int @ A2 @ ( insert_set_int @ A2 @ B2 ) ) ).
% insertI1
thf(fact_377_insertI1,axiom,
! [A2: int,B2: set_int] : ( member_int @ A2 @ ( insert_int @ A2 @ B2 ) ) ).
% insertI1
thf(fact_378_insertI1,axiom,
! [A2: nat,B2: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_379_insertE,axiom,
! [A2: set_int,B: set_int,A: set_set_int] :
( ( member_set_int @ A2 @ ( insert_set_int @ B @ A ) )
=> ( ( A2 != B )
=> ( member_set_int @ A2 @ A ) ) ) ).
% insertE
thf(fact_380_insertE,axiom,
! [A2: int,B: int,A: set_int] :
( ( member_int @ A2 @ ( insert_int @ B @ A ) )
=> ( ( A2 != B )
=> ( member_int @ A2 @ A ) ) ) ).
% insertE
thf(fact_381_insertE,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_382_mult_Oleft__commute,axiom,
! [B: int,A2: int,C2: int] :
( ( times_times_int @ B @ ( times_times_int @ A2 @ C2 ) )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_383_mult_Oleft__commute,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A2 @ C2 ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.left_commute
thf(fact_384_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_385_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_386_mult_Oassoc,axiom,
! [A2: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C2 )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_387_mult_Oassoc,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C2 )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C2 ) ) ) ).
% mult.assoc
thf(fact_388_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: int,B: int,C2: int] :
( ( times_times_int @ ( times_times_int @ A2 @ B ) @ C2 )
= ( times_times_int @ A2 @ ( times_times_int @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_389_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C2 )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C2 ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_390_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
% int_distrib(2)
thf(fact_391_int__distrib_I1_J,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
% int_distrib(1)
thf(fact_392_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_393_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_394_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_395_mult__right__cancel,axiom,
! [C2: int,A2: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ A2 @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_396_mult__right__cancel,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_397_mult__left__cancel,axiom,
! [C2: int,A2: int,B: int] :
( ( C2 != zero_zero_int )
=> ( ( ( times_times_int @ C2 @ A2 )
= ( times_times_int @ C2 @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_398_mult__left__cancel,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( C2 != zero_zero_nat )
=> ( ( ( times_times_nat @ C2 @ A2 )
= ( times_times_nat @ C2 @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_399_no__zero__divisors,axiom,
! [A2: int,B: int] :
( ( A2 != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A2 @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_400_no__zero__divisors,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_401_divisors__zero,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
=> ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_402_divisors__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_403_mult__not__zero,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
!= zero_zero_int )
=> ( ( A2 != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_404_mult__not__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_405_combine__common__factor,axiom,
! [A2: int,E: int,B: int,C2: int] :
( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_406_combine__common__factor,axiom,
! [A2: nat,E: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ E ) @ C2 ) ) ).
% combine_common_factor
thf(fact_407_distrib__right,axiom,
! [A2: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_408_distrib__right,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% distrib_right
thf(fact_409_distrib__left,axiom,
! [A2: int,B: int,C2: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C2 ) ) ) ).
% distrib_left
thf(fact_410_distrib__left,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( times_times_nat @ A2 @ ( plus_plus_nat @ B @ C2 ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C2 ) ) ) ).
% distrib_left
thf(fact_411_comm__semiring__class_Odistrib,axiom,
! [A2: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_412_comm__semiring__class_Odistrib,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).
% comm_semiring_class.distrib
thf(fact_413_ring__class_Oring__distribs_I1_J,axiom,
! [A2: int,B: int,C2: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B @ C2 ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B ) @ ( times_times_int @ A2 @ C2 ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_414_ring__class_Oring__distribs_I2_J,axiom,
! [A2: int,B: int,C2: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B ) @ C2 )
= ( plus_plus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_415_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_416_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_417_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_418_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_419_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_420_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_421_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_422_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_423_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_424_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_425_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_426_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_427_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_428_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_429_order__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_430_order__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_431_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_432_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_433_order__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_434_order__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_435_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_436_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_437_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_438_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_439_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_440_ord__less__eq__subst,axiom,
! [A2: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_441_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_442_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B: nat,C2: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_443_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B: int,C2: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_444_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B: int,C2: int] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_445_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_446_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_447_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_448_order__less__asym_H,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_449_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_450_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_451_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_452_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_453_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_454_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_455_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_456_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_457_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_458_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_459_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_460_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_461_dual__order_Ostrict__trans,axiom,
! [B: int,A2: int,C2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_462_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_463_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_464_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_465_order_Ostrict__trans,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_466_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B5: nat] :
( ( ord_less_nat @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B5: nat] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_467_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B: int] :
( ! [A4: int,B5: int] :
( ( ord_less_int @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B5: int] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_468_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N4: nat] :
( ( P3 @ N4 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ~ ( P3 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_469_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_470_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_471_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_472_dual__order_Oasym,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ~ ( ord_less_int @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_473_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_474_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_475_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_476_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_477_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X2: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X2 )
=> ( P @ Y3 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_478_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_479_ord__less__eq__trans,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( B = C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_480_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_481_ord__eq__less__trans,axiom,
! [A2: int,B: int,C2: int] :
( ( A2 = B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_482_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_483_order_Oasym,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ).
% order.asym
thf(fact_484_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_485_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_486_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_487_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_488_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_489_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_490_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_491_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_492_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_493_mult__less__cancel__right__disj,axiom,
! [A2: int,C2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A2 @ B ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_494_mult__strict__right__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_495_mult__strict__right__mono,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono
thf(fact_496_mult__strict__right__mono__neg,axiom,
! [B: int,A2: int,C2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_497_mult__less__cancel__left__disj,axiom,
! [C2: int,A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C2 )
& ( ord_less_int @ A2 @ B ) )
| ( ( ord_less_int @ C2 @ zero_zero_int )
& ( ord_less_int @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_498_mult__strict__left__mono,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C2 )
=> ( ord_less_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_499_mult__strict__left__mono,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_500_mult__strict__left__mono__neg,axiom,
! [B: int,A2: int,C2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_501_mult__less__cancel__left__pos,axiom,
! [C2: int,A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C2 )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_int @ A2 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_502_mult__less__cancel__left__neg,axiom,
! [C2: int,A2: int,B: int] :
( ( ord_less_int @ C2 @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B ) )
= ( ord_less_int @ B @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_503_zero__less__mult__pos2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_504_zero__less__mult__pos2,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_505_zero__less__mult__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_506_zero__less__mult__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_507_zero__less__mult__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_508_mult__pos__neg2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_509_mult__pos__neg2,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_510_mult__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_511_mult__pos__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_512_mult__pos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_513_mult__pos__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_514_mult__neg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_515_mult__neg__pos,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_516_mult__less__0__iff,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_517_not__square__less__zero,axiom,
! [A2: int] :
~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_518_mult__neg__neg,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_519_int__if,axiom,
! [P: $o,A2: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_520_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A3: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_521_singleton__inject,axiom,
! [A2: nat,B: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_522_singleton__inject,axiom,
! [A2: int,B: int] :
( ( ( insert_int @ A2 @ bot_bot_set_int )
= ( insert_int @ B @ bot_bot_set_int ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_523_singleton__inject,axiom,
! [A2: set_int,B: set_int] :
( ( ( insert_set_int @ A2 @ bot_bot_set_set_int )
= ( insert_set_int @ B @ bot_bot_set_set_int ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_524_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_525_insert__not__empty,axiom,
! [A2: int,A: set_int] :
( ( insert_int @ A2 @ A )
!= bot_bot_set_int ) ).
% insert_not_empty
thf(fact_526_insert__not__empty,axiom,
! [A2: set_int,A: set_set_int] :
( ( insert_set_int @ A2 @ A )
!= bot_bot_set_set_int ) ).
% insert_not_empty
thf(fact_527_doubleton__eq__iff,axiom,
! [A2: nat,B: nat,C2: nat,D2: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C2 @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C2 )
& ( B = D2 ) )
| ( ( A2 = D2 )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_528_doubleton__eq__iff,axiom,
! [A2: int,B: int,C2: int,D2: int] :
( ( ( insert_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) )
= ( insert_int @ C2 @ ( insert_int @ D2 @ bot_bot_set_int ) ) )
= ( ( ( A2 = C2 )
& ( B = D2 ) )
| ( ( A2 = D2 )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_529_doubleton__eq__iff,axiom,
! [A2: set_int,B: set_int,C2: set_int,D2: set_int] :
( ( ( insert_set_int @ A2 @ ( insert_set_int @ B @ bot_bot_set_set_int ) )
= ( insert_set_int @ C2 @ ( insert_set_int @ D2 @ bot_bot_set_set_int ) ) )
= ( ( ( A2 = C2 )
& ( B = D2 ) )
| ( ( A2 = D2 )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_530_singleton__iff,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_531_singleton__iff,axiom,
! [B: int,A2: int] :
( ( member_int @ B @ ( insert_int @ A2 @ bot_bot_set_int ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_532_singleton__iff,axiom,
! [B: set_int,A2: set_int] :
( ( member_set_int @ B @ ( insert_set_int @ A2 @ bot_bot_set_set_int ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_533_singletonD,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_534_singletonD,axiom,
! [B: int,A2: int] :
( ( member_int @ B @ ( insert_int @ A2 @ bot_bot_set_int ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_535_singletonD,axiom,
! [B: set_int,A2: set_int] :
( ( member_set_int @ B @ ( insert_set_int @ A2 @ bot_bot_set_set_int ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_536_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_537_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_538_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_539_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_540_verit__sum__simplify,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% verit_sum_simplify
thf(fact_541_bot_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_542_bot_Onot__eq__extremum,axiom,
! [A2: set_int] :
( ( A2 != bot_bot_set_int )
= ( ord_less_set_int @ bot_bot_set_int @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_543_bot_Onot__eq__extremum,axiom,
! [A2: set_set_int] :
( ( A2 != bot_bot_set_set_int )
= ( ord_less_set_set_int @ bot_bot_set_set_int @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_544_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_545_bot_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_546_bot_Oextremum__strict,axiom,
! [A2: set_int] :
~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% bot.extremum_strict
thf(fact_547_bot_Oextremum__strict,axiom,
! [A2: set_set_int] :
~ ( ord_less_set_set_int @ A2 @ bot_bot_set_set_int ) ).
% bot.extremum_strict
thf(fact_548_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_549_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_550_I__def,axiom,
( i
= ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ ( semiri1314217659103216013at_int @ n ) @ bot_bot_set_int ) ) ) ).
% I_def
thf(fact_551_mult__cancel__right2,axiom,
! [A2: int,C2: int] :
( ( ( times_times_int @ A2 @ C2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_552_mult__cancel__right1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_553_mult__cancel__left2,axiom,
! [C2: int,A2: int] :
( ( ( times_times_int @ C2 @ A2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_554_mult__cancel__left1,axiom,
! [C2: int,B: int] :
( ( C2
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_555_bij__betw__add,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ ( plus_plus_nat @ A2 ) @ A @ B2 )
= ( ( image_nat_nat @ ( plus_plus_nat @ A2 ) @ A )
= B2 ) ) ).
% bij_betw_add
thf(fact_556_bij__betw__add,axiom,
! [A2: int,A: set_int,B2: set_int] :
( ( bij_betw_int_int @ ( plus_plus_int @ A2 ) @ A @ B2 )
= ( ( image_int_int @ ( plus_plus_int @ A2 ) @ A )
= B2 ) ) ).
% bij_betw_add
thf(fact_557_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_558_mult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% mult_1
thf(fact_559_mult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% mult_1
thf(fact_560_mult_Oright__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.right_neutral
thf(fact_561_mult_Oright__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_562_UNIV__I,axiom,
! [X: set_int] : ( member_set_int @ X @ top_top_set_set_int ) ).
% UNIV_I
thf(fact_563_UNIV__I,axiom,
! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% UNIV_I
thf(fact_564_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_565_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_566_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_567_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_568_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_569_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_570_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_571_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_572_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_573_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_574_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_575_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_576_mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_577_mult__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_578_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_579_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_580_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times_nat @ M2 @ N ) )
=> ( ( N = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_581_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_582_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_583_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_584_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_585_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_586_add__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M2 @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_587_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_588_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_589_UNIV__eq__I,axiom,
! [A: set_set_int] :
( ! [X2: set_int] : ( member_set_int @ X2 @ A )
=> ( top_top_set_set_int = A ) ) ).
% UNIV_eq_I
thf(fact_590_UNIV__eq__I,axiom,
! [A: set_int] :
( ! [X2: int] : ( member_int @ X2 @ A )
=> ( top_top_set_int = A ) ) ).
% UNIV_eq_I
thf(fact_591_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_592_UNIV__witness,axiom,
? [X2: set_int] : ( member_set_int @ X2 @ top_top_set_set_int ) ).
% UNIV_witness
thf(fact_593_UNIV__witness,axiom,
? [X2: int] : ( member_int @ X2 @ top_top_set_int ) ).
% UNIV_witness
thf(fact_594_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_595_surj__def,axiom,
! [F: set_int > int] :
( ( ( image_set_int_int @ F @ top_top_set_set_int )
= top_top_set_int )
= ( ! [Y5: int] :
? [X3: set_int] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_596_surj__def,axiom,
! [F: int > set_int] :
( ( ( image_int_set_int @ F @ top_top_set_int )
= top_top_set_set_int )
= ( ! [Y5: set_int] :
? [X3: int] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_597_surj__def,axiom,
! [F: int > int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
= ( ! [Y5: int] :
? [X3: int] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_598_surj__def,axiom,
! [F: int > nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
= ( ! [Y5: nat] :
? [X3: int] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_599_surj__def,axiom,
! [F: nat > set_int] :
( ( ( image_nat_set_int @ F @ top_top_set_nat )
= top_top_set_set_int )
= ( ! [Y5: set_int] :
? [X3: nat] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_600_surj__def,axiom,
! [F: nat > int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
= ( ! [Y5: int] :
? [X3: nat] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_601_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y5: nat] :
? [X3: nat] :
( Y5
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_602_surjI,axiom,
! [G: set_int > int,F: int > set_int] :
( ! [X2: int] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_set_int_int @ G @ top_top_set_set_int )
= top_top_set_int ) ) ).
% surjI
thf(fact_603_surjI,axiom,
! [G: int > set_int,F: set_int > int] :
( ! [X2: set_int] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_int_set_int @ G @ top_top_set_int )
= top_top_set_set_int ) ) ).
% surjI
thf(fact_604_surjI,axiom,
! [G: int > int,F: int > int] :
( ! [X2: int] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_int_int @ G @ top_top_set_int )
= top_top_set_int ) ) ).
% surjI
thf(fact_605_surjI,axiom,
! [G: int > nat,F: nat > int] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_int_nat @ G @ top_top_set_int )
= top_top_set_nat ) ) ).
% surjI
thf(fact_606_surjI,axiom,
! [G: nat > set_int,F: set_int > nat] :
( ! [X2: set_int] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_set_int @ G @ top_top_set_nat )
= top_top_set_set_int ) ) ).
% surjI
thf(fact_607_surjI,axiom,
! [G: nat > int,F: int > nat] :
( ! [X2: int] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_int @ G @ top_top_set_nat )
= top_top_set_int ) ) ).
% surjI
thf(fact_608_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_609_surjE,axiom,
! [F: set_int > int,Y: int] :
( ( ( image_set_int_int @ F @ top_top_set_set_int )
= top_top_set_int )
=> ~ ! [X2: set_int] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_610_surjE,axiom,
! [F: int > set_int,Y: set_int] :
( ( ( image_int_set_int @ F @ top_top_set_int )
= top_top_set_set_int )
=> ~ ! [X2: int] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_611_surjE,axiom,
! [F: int > int,Y: int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ~ ! [X2: int] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_612_surjE,axiom,
! [F: int > nat,Y: nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
=> ~ ! [X2: int] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_613_surjE,axiom,
! [F: nat > set_int,Y: set_int] :
( ( ( image_nat_set_int @ F @ top_top_set_nat )
= top_top_set_set_int )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_614_surjE,axiom,
! [F: nat > int,Y: int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_615_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_616_surjD,axiom,
! [F: set_int > int,Y: int] :
( ( ( image_set_int_int @ F @ top_top_set_set_int )
= top_top_set_int )
=> ? [X2: set_int] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_617_surjD,axiom,
! [F: int > set_int,Y: set_int] :
( ( ( image_int_set_int @ F @ top_top_set_int )
= top_top_set_set_int )
=> ? [X2: int] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_618_surjD,axiom,
! [F: int > int,Y: int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ? [X2: int] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_619_surjD,axiom,
! [F: int > nat,Y: nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
=> ? [X2: int] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_620_surjD,axiom,
! [F: nat > set_int,Y: set_int] :
( ( ( image_nat_set_int @ F @ top_top_set_nat )
= top_top_set_set_int )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_621_surjD,axiom,
! [F: nat > int,Y: int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_622_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_623_bij__iff,axiom,
! [F: int > int] :
( ( bij_betw_int_int @ F @ top_top_set_int @ top_top_set_int )
= ( ! [X3: int] :
? [Y5: int] :
( ( ( F @ Y5 )
= X3 )
& ! [Z4: int] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y5 ) ) ) ) ) ).
% bij_iff
thf(fact_624_bij__iff,axiom,
! [F: int > nat] :
( ( bij_betw_int_nat @ F @ top_top_set_int @ top_top_set_nat )
= ( ! [X3: nat] :
? [Y5: int] :
( ( ( F @ Y5 )
= X3 )
& ! [Z4: int] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y5 ) ) ) ) ) ).
% bij_iff
thf(fact_625_bij__iff,axiom,
! [F: nat > set_int] :
( ( bij_betw_nat_set_int @ F @ top_top_set_nat @ top_top_set_set_int )
= ( ! [X3: set_int] :
? [Y5: nat] :
( ( ( F @ Y5 )
= X3 )
& ! [Z4: nat] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y5 ) ) ) ) ) ).
% bij_iff
thf(fact_626_bij__iff,axiom,
! [F: nat > int] :
( ( bij_betw_nat_int @ F @ top_top_set_nat @ top_top_set_int )
= ( ! [X3: int] :
? [Y5: nat] :
( ( ( F @ Y5 )
= X3 )
& ! [Z4: nat] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y5 ) ) ) ) ) ).
% bij_iff
thf(fact_627_bij__iff,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
= ( ! [X3: nat] :
? [Y5: nat] :
( ( ( F @ Y5 )
= X3 )
& ! [Z4: nat] :
( ( ( F @ Z4 )
= X3 )
=> ( Z4 = Y5 ) ) ) ) ) ).
% bij_iff
thf(fact_628_bij__pointE,axiom,
! [F: int > int,Y: int] :
( ( bij_betw_int_int @ F @ top_top_set_int @ top_top_set_int )
=> ~ ! [X2: int] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X6: int] :
( ( Y
= ( F @ X6 ) )
=> ( X6 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_629_bij__pointE,axiom,
! [F: int > nat,Y: nat] :
( ( bij_betw_int_nat @ F @ top_top_set_int @ top_top_set_nat )
=> ~ ! [X2: int] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X6: int] :
( ( Y
= ( F @ X6 ) )
=> ( X6 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_630_bij__pointE,axiom,
! [F: nat > set_int,Y: set_int] :
( ( bij_betw_nat_set_int @ F @ top_top_set_nat @ top_top_set_set_int )
=> ~ ! [X2: nat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X6: nat] :
( ( Y
= ( F @ X6 ) )
=> ( X6 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_631_bij__pointE,axiom,
! [F: nat > int,Y: int] :
( ( bij_betw_nat_int @ F @ top_top_set_nat @ top_top_set_int )
=> ~ ! [X2: nat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X6: nat] :
( ( Y
= ( F @ X6 ) )
=> ( X6 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_632_bij__pointE,axiom,
! [F: nat > nat,Y: nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ~ ! [X2: nat] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X6: nat] :
( ( Y
= ( F @ X6 ) )
=> ( X6 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_633_involuntory__imp__bij,axiom,
! [F: int > int] :
( ! [X2: int] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_betw_int_int @ F @ top_top_set_int @ top_top_set_int ) ) ).
% involuntory_imp_bij
thf(fact_634_involuntory__imp__bij,axiom,
! [F: nat > nat] :
( ! [X2: nat] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat ) ) ).
% involuntory_imp_bij
thf(fact_635_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_636_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_637_int__ops_I7_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_638_bij__betwE,axiom,
! [F: nat > set_int,A: set_nat,B2: set_set_int] :
( ( bij_betw_nat_set_int @ F @ A @ B2 )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_set_int @ ( F @ X4 ) @ B2 ) ) ) ).
% bij_betwE
thf(fact_639_bij__betw__inv,axiom,
! [F: set_int > nat,A: set_set_int,B2: set_nat] :
( ( bij_betw_set_int_nat @ F @ A @ B2 )
=> ? [G2: nat > set_int] : ( bij_betw_nat_set_int @ G2 @ B2 @ A ) ) ).
% bij_betw_inv
thf(fact_640_bij__betw__inv,axiom,
! [F: nat > set_int,A: set_nat,B2: set_set_int] :
( ( bij_betw_nat_set_int @ F @ A @ B2 )
=> ? [G2: set_int > nat] : ( bij_betw_set_int_nat @ G2 @ B2 @ A ) ) ).
% bij_betw_inv
thf(fact_641_bij__betw__ball,axiom,
! [F: nat > set_int,A: set_nat,B2: set_set_int,Phi: set_int > $o] :
( ( bij_betw_nat_set_int @ F @ A @ B2 )
=> ( ( ! [X3: set_int] :
( ( member_set_int @ X3 @ B2 )
=> ( Phi @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( Phi @ ( F @ X3 ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_642_bij__betw__cong,axiom,
! [A: set_nat,F: nat > set_int,G: nat > set_int,A5: set_set_int] :
( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( bij_betw_nat_set_int @ F @ A @ A5 )
= ( bij_betw_nat_set_int @ G @ A @ A5 ) ) ) ).
% bij_betw_cong
thf(fact_643_bij__betw__apply,axiom,
! [F: set_int > set_int,A: set_set_int,B2: set_set_int,A2: set_int] :
( ( bij_be5268973184346298300et_int @ F @ A @ B2 )
=> ( ( member_set_int @ A2 @ A )
=> ( member_set_int @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_644_bij__betw__apply,axiom,
! [F: set_int > int,A: set_set_int,B2: set_int,A2: set_int] :
( ( bij_betw_set_int_int @ F @ A @ B2 )
=> ( ( member_set_int @ A2 @ A )
=> ( member_int @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_645_bij__betw__apply,axiom,
! [F: set_int > nat,A: set_set_int,B2: set_nat,A2: set_int] :
( ( bij_betw_set_int_nat @ F @ A @ B2 )
=> ( ( member_set_int @ A2 @ A )
=> ( member_nat @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_646_bij__betw__apply,axiom,
! [F: int > set_int,A: set_int,B2: set_set_int,A2: int] :
( ( bij_betw_int_set_int @ F @ A @ B2 )
=> ( ( member_int @ A2 @ A )
=> ( member_set_int @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_647_bij__betw__apply,axiom,
! [F: int > int,A: set_int,B2: set_int,A2: int] :
( ( bij_betw_int_int @ F @ A @ B2 )
=> ( ( member_int @ A2 @ A )
=> ( member_int @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_648_bij__betw__apply,axiom,
! [F: int > nat,A: set_int,B2: set_nat,A2: int] :
( ( bij_betw_int_nat @ F @ A @ B2 )
=> ( ( member_int @ A2 @ A )
=> ( member_nat @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_649_bij__betw__apply,axiom,
! [F: nat > int,A: set_nat,B2: set_int,A2: nat] :
( ( bij_betw_nat_int @ F @ A @ B2 )
=> ( ( member_nat @ A2 @ A )
=> ( member_int @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_650_bij__betw__apply,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat,A2: nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( ( member_nat @ A2 @ A )
=> ( member_nat @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_651_bij__betw__apply,axiom,
! [F: nat > set_int,A: set_nat,B2: set_set_int,A2: nat] :
( ( bij_betw_nat_set_int @ F @ A @ B2 )
=> ( ( member_nat @ A2 @ A )
=> ( member_set_int @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_652_bij__betw__iff__bijections,axiom,
( bij_be5268973184346298300et_int
= ( ^ [F2: set_int > set_int,A6: set_set_int,B6: set_set_int] :
? [G3: set_int > set_int] :
( ! [X3: set_int] :
( ( member_set_int @ X3 @ A6 )
=> ( ( member_set_int @ ( F2 @ X3 ) @ B6 )
& ( ( G3 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: set_int] :
( ( member_set_int @ X3 @ B6 )
=> ( ( member_set_int @ ( G3 @ X3 ) @ A6 )
& ( ( F2 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_653_bij__betw__iff__bijections,axiom,
( bij_betw_int_set_int
= ( ^ [F2: int > set_int,A6: set_int,B6: set_set_int] :
? [G3: set_int > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A6 )
=> ( ( member_set_int @ ( F2 @ X3 ) @ B6 )
& ( ( G3 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: set_int] :
( ( member_set_int @ X3 @ B6 )
=> ( ( member_int @ ( G3 @ X3 ) @ A6 )
& ( ( F2 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_654_bij__betw__iff__bijections,axiom,
( bij_betw_set_int_int
= ( ^ [F2: set_int > int,A6: set_set_int,B6: set_int] :
? [G3: int > set_int] :
( ! [X3: set_int] :
( ( member_set_int @ X3 @ A6 )
=> ( ( member_int @ ( F2 @ X3 ) @ B6 )
& ( ( G3 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: int] :
( ( member_int @ X3 @ B6 )
=> ( ( member_set_int @ ( G3 @ X3 ) @ A6 )
& ( ( F2 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_655_bij__betw__iff__bijections,axiom,
( bij_betw_int_int
= ( ^ [F2: int > int,A6: set_int,B6: set_int] :
? [G3: int > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A6 )
=> ( ( member_int @ ( F2 @ X3 ) @ B6 )
& ( ( G3 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: int] :
( ( member_int @ X3 @ B6 )
=> ( ( member_int @ ( G3 @ X3 ) @ A6 )
& ( ( F2 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_656_bij__betw__iff__bijections,axiom,
( bij_betw_nat_int
= ( ^ [F2: nat > int,A6: set_nat,B6: set_int] :
? [G3: int > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( ( member_int @ ( F2 @ X3 ) @ B6 )
& ( ( G3 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: int] :
( ( member_int @ X3 @ B6 )
=> ( ( member_nat @ ( G3 @ X3 ) @ A6 )
& ( ( F2 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_657_bij__betw__iff__bijections,axiom,
( bij_betw_set_int_nat
= ( ^ [F2: set_int > nat,A6: set_set_int,B6: set_nat] :
? [G3: nat > set_int] :
( ! [X3: set_int] :
( ( member_set_int @ X3 @ A6 )
=> ( ( member_nat @ ( F2 @ X3 ) @ B6 )
& ( ( G3 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B6 )
=> ( ( member_set_int @ ( G3 @ X3 ) @ A6 )
& ( ( F2 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_658_bij__betw__iff__bijections,axiom,
( bij_betw_int_nat
= ( ^ [F2: int > nat,A6: set_int,B6: set_nat] :
? [G3: nat > int] :
( ! [X3: int] :
( ( member_int @ X3 @ A6 )
=> ( ( member_nat @ ( F2 @ X3 ) @ B6 )
& ( ( G3 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B6 )
=> ( ( member_int @ ( G3 @ X3 ) @ A6 )
& ( ( F2 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_659_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat
= ( ^ [F2: nat > nat,A6: set_nat,B6: set_nat] :
? [G3: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( ( member_nat @ ( F2 @ X3 ) @ B6 )
& ( ( G3 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B6 )
=> ( ( member_nat @ ( G3 @ X3 ) @ A6 )
& ( ( F2 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_660_bij__betw__iff__bijections,axiom,
( bij_betw_nat_set_int
= ( ^ [F2: nat > set_int,A6: set_nat,B6: set_set_int] :
? [G3: set_int > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A6 )
=> ( ( member_set_int @ ( F2 @ X3 ) @ B6 )
& ( ( G3 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: set_int] :
( ( member_set_int @ X3 @ B6 )
=> ( ( member_nat @ ( G3 @ X3 ) @ A6 )
& ( ( F2 @ ( G3 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_661_bij__plus,axiom,
! [A2: int] : ( bij_betw_int_int @ ( plus_plus_int @ A2 ) @ top_top_set_int @ top_top_set_int ) ).
% bij_plus
thf(fact_662_bij__is__surj,axiom,
! [F: set_int > int] :
( ( bij_betw_set_int_int @ F @ top_top_set_set_int @ top_top_set_int )
=> ( ( image_set_int_int @ F @ top_top_set_set_int )
= top_top_set_int ) ) ).
% bij_is_surj
thf(fact_663_bij__is__surj,axiom,
! [F: int > set_int] :
( ( bij_betw_int_set_int @ F @ top_top_set_int @ top_top_set_set_int )
=> ( ( image_int_set_int @ F @ top_top_set_int )
= top_top_set_set_int ) ) ).
% bij_is_surj
thf(fact_664_bij__is__surj,axiom,
! [F: int > int] :
( ( bij_betw_int_int @ F @ top_top_set_int @ top_top_set_int )
=> ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int ) ) ).
% bij_is_surj
thf(fact_665_bij__is__surj,axiom,
! [F: int > nat] :
( ( bij_betw_int_nat @ F @ top_top_set_int @ top_top_set_nat )
=> ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat ) ) ).
% bij_is_surj
thf(fact_666_bij__is__surj,axiom,
! [F: nat > set_int] :
( ( bij_betw_nat_set_int @ F @ top_top_set_nat @ top_top_set_set_int )
=> ( ( image_nat_set_int @ F @ top_top_set_nat )
= top_top_set_set_int ) ) ).
% bij_is_surj
thf(fact_667_bij__is__surj,axiom,
! [F: nat > int] :
( ( bij_betw_nat_int @ F @ top_top_set_nat @ top_top_set_int )
=> ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int ) ) ).
% bij_is_surj
thf(fact_668_bij__is__surj,axiom,
! [F: nat > nat] :
( ( bij_betw_nat_nat @ F @ top_top_set_nat @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bij_is_surj
thf(fact_669_bij__betw__imp__surj,axiom,
! [F: int > set_int,A: set_int] :
( ( bij_betw_int_set_int @ F @ A @ top_top_set_set_int )
=> ( ( image_int_set_int @ F @ top_top_set_int )
= top_top_set_set_int ) ) ).
% bij_betw_imp_surj
thf(fact_670_bij__betw__imp__surj,axiom,
! [F: nat > set_int,A: set_nat] :
( ( bij_betw_nat_set_int @ F @ A @ top_top_set_set_int )
=> ( ( image_nat_set_int @ F @ top_top_set_nat )
= top_top_set_set_int ) ) ).
% bij_betw_imp_surj
thf(fact_671_bij__betw__imp__surj,axiom,
! [F: set_int > int,A: set_set_int] :
( ( bij_betw_set_int_int @ F @ A @ top_top_set_int )
=> ( ( image_set_int_int @ F @ top_top_set_set_int )
= top_top_set_int ) ) ).
% bij_betw_imp_surj
thf(fact_672_bij__betw__imp__surj,axiom,
! [F: int > int,A: set_int] :
( ( bij_betw_int_int @ F @ A @ top_top_set_int )
=> ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int ) ) ).
% bij_betw_imp_surj
thf(fact_673_bij__betw__imp__surj,axiom,
! [F: nat > int,A: set_nat] :
( ( bij_betw_nat_int @ F @ A @ top_top_set_int )
=> ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int ) ) ).
% bij_betw_imp_surj
thf(fact_674_bij__betw__imp__surj,axiom,
! [F: int > nat,A: set_int] :
( ( bij_betw_int_nat @ F @ A @ top_top_set_nat )
=> ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat ) ) ).
% bij_betw_imp_surj
thf(fact_675_bij__betw__imp__surj,axiom,
! [F: nat > nat,A: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ).
% bij_betw_imp_surj
thf(fact_676_top_Oextremum__strict,axiom,
! [A2: set_int] :
~ ( ord_less_set_int @ top_top_set_int @ A2 ) ).
% top.extremum_strict
thf(fact_677_top_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).
% top.extremum_strict
thf(fact_678_top_Onot__eq__extremum,axiom,
! [A2: set_int] :
( ( A2 != top_top_set_int )
= ( ord_less_set_int @ A2 @ top_top_set_int ) ) ).
% top.not_eq_extremum
thf(fact_679_top_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != top_top_set_nat )
= ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_680_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_681_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_682_comm__monoid__mult__class_Omult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_683_comm__monoid__mult__class_Omult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_684_mult_Ocomm__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.comm_neutral
thf(fact_685_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_686_empty__not__UNIV,axiom,
bot_bot_set_set_int != top_top_set_set_int ).
% empty_not_UNIV
thf(fact_687_empty__not__UNIV,axiom,
bot_bot_set_int != top_top_set_int ).
% empty_not_UNIV
thf(fact_688_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_689_range__eqI,axiom,
! [B: int,F: set_int > int,X: set_int] :
( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_set_int_int @ F @ top_top_set_set_int ) ) ) ).
% range_eqI
thf(fact_690_range__eqI,axiom,
! [B: set_int,F: int > set_int,X: int] :
( ( B
= ( F @ X ) )
=> ( member_set_int @ B @ ( image_int_set_int @ F @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_691_range__eqI,axiom,
! [B: int,F: int > int,X: int] :
( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_int_int @ F @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_692_range__eqI,axiom,
! [B: nat,F: int > nat,X: int] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_int_nat @ F @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_693_range__eqI,axiom,
! [B: set_int,F: nat > set_int,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_set_int @ B @ ( image_nat_set_int @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_694_range__eqI,axiom,
! [B: int,F: nat > int,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_int @ B @ ( image_nat_int @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_695_range__eqI,axiom,
! [B: nat,F: nat > nat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_696_rangeI,axiom,
! [F: set_int > int,X: set_int] : ( member_int @ ( F @ X ) @ ( image_set_int_int @ F @ top_top_set_set_int ) ) ).
% rangeI
thf(fact_697_rangeI,axiom,
! [F: int > set_int,X: int] : ( member_set_int @ ( F @ X ) @ ( image_int_set_int @ F @ top_top_set_int ) ) ).
% rangeI
thf(fact_698_rangeI,axiom,
! [F: int > int,X: int] : ( member_int @ ( F @ X ) @ ( image_int_int @ F @ top_top_set_int ) ) ).
% rangeI
thf(fact_699_rangeI,axiom,
! [F: int > nat,X: int] : ( member_nat @ ( F @ X ) @ ( image_int_nat @ F @ top_top_set_int ) ) ).
% rangeI
thf(fact_700_rangeI,axiom,
! [F: nat > set_int,X: nat] : ( member_set_int @ ( F @ X ) @ ( image_nat_set_int @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_701_rangeI,axiom,
! [F: nat > int,X: nat] : ( member_int @ ( F @ X ) @ ( image_nat_int @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_702_rangeI,axiom,
! [F: nat > nat,X: nat] : ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_703_insert__UNIV,axiom,
! [X: set_int] :
( ( insert_set_int @ X @ top_top_set_set_int )
= top_top_set_set_int ) ).
% insert_UNIV
thf(fact_704_insert__UNIV,axiom,
! [X: int] :
( ( insert_int @ X @ top_top_set_int )
= top_top_set_int ) ).
% insert_UNIV
thf(fact_705_insert__UNIV,axiom,
! [X: nat] :
( ( insert_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_706_bij__betw__empty2,axiom,
! [F: nat > nat,A: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bij_betw_empty2
thf(fact_707_bij__betw__empty2,axiom,
! [F: int > nat,A: set_int] :
( ( bij_betw_int_nat @ F @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_int ) ) ).
% bij_betw_empty2
thf(fact_708_bij__betw__empty2,axiom,
! [F: set_int > nat,A: set_set_int] :
( ( bij_betw_set_int_nat @ F @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_set_int ) ) ).
% bij_betw_empty2
thf(fact_709_bij__betw__empty2,axiom,
! [F: nat > int,A: set_nat] :
( ( bij_betw_nat_int @ F @ A @ bot_bot_set_int )
=> ( A = bot_bot_set_nat ) ) ).
% bij_betw_empty2
thf(fact_710_bij__betw__empty2,axiom,
! [F: int > int,A: set_int] :
( ( bij_betw_int_int @ F @ A @ bot_bot_set_int )
=> ( A = bot_bot_set_int ) ) ).
% bij_betw_empty2
thf(fact_711_bij__betw__empty2,axiom,
! [F: set_int > int,A: set_set_int] :
( ( bij_betw_set_int_int @ F @ A @ bot_bot_set_int )
=> ( A = bot_bot_set_set_int ) ) ).
% bij_betw_empty2
thf(fact_712_bij__betw__empty2,axiom,
! [F: nat > set_int,A: set_nat] :
( ( bij_betw_nat_set_int @ F @ A @ bot_bot_set_set_int )
=> ( A = bot_bot_set_nat ) ) ).
% bij_betw_empty2
thf(fact_713_bij__betw__empty2,axiom,
! [F: int > set_int,A: set_int] :
( ( bij_betw_int_set_int @ F @ A @ bot_bot_set_set_int )
=> ( A = bot_bot_set_int ) ) ).
% bij_betw_empty2
thf(fact_714_bij__betw__empty2,axiom,
! [F: set_int > set_int,A: set_set_int] :
( ( bij_be5268973184346298300et_int @ F @ A @ bot_bot_set_set_int )
=> ( A = bot_bot_set_set_int ) ) ).
% bij_betw_empty2
thf(fact_715_bij__betw__empty1,axiom,
! [F: nat > nat,A: set_nat] :
( ( bij_betw_nat_nat @ F @ bot_bot_set_nat @ A )
=> ( A = bot_bot_set_nat ) ) ).
% bij_betw_empty1
thf(fact_716_bij__betw__empty1,axiom,
! [F: nat > int,A: set_int] :
( ( bij_betw_nat_int @ F @ bot_bot_set_nat @ A )
=> ( A = bot_bot_set_int ) ) ).
% bij_betw_empty1
thf(fact_717_bij__betw__empty1,axiom,
! [F: nat > set_int,A: set_set_int] :
( ( bij_betw_nat_set_int @ F @ bot_bot_set_nat @ A )
=> ( A = bot_bot_set_set_int ) ) ).
% bij_betw_empty1
thf(fact_718_bij__betw__empty1,axiom,
! [F: int > nat,A: set_nat] :
( ( bij_betw_int_nat @ F @ bot_bot_set_int @ A )
=> ( A = bot_bot_set_nat ) ) ).
% bij_betw_empty1
thf(fact_719_bij__betw__empty1,axiom,
! [F: int > int,A: set_int] :
( ( bij_betw_int_int @ F @ bot_bot_set_int @ A )
=> ( A = bot_bot_set_int ) ) ).
% bij_betw_empty1
thf(fact_720_bij__betw__empty1,axiom,
! [F: int > set_int,A: set_set_int] :
( ( bij_betw_int_set_int @ F @ bot_bot_set_int @ A )
=> ( A = bot_bot_set_set_int ) ) ).
% bij_betw_empty1
thf(fact_721_bij__betw__empty1,axiom,
! [F: set_int > nat,A: set_nat] :
( ( bij_betw_set_int_nat @ F @ bot_bot_set_set_int @ A )
=> ( A = bot_bot_set_nat ) ) ).
% bij_betw_empty1
thf(fact_722_bij__betw__empty1,axiom,
! [F: set_int > int,A: set_int] :
( ( bij_betw_set_int_int @ F @ bot_bot_set_set_int @ A )
=> ( A = bot_bot_set_int ) ) ).
% bij_betw_empty1
thf(fact_723_bij__betw__empty1,axiom,
! [F: set_int > set_int,A: set_set_int] :
( ( bij_be5268973184346298300et_int @ F @ bot_bot_set_set_int @ A )
=> ( A = bot_bot_set_set_int ) ) ).
% bij_betw_empty1
thf(fact_724_bij__betw__imp__surj__on,axiom,
! [F: int > int,A: set_int,B2: set_int] :
( ( bij_betw_int_int @ F @ A @ B2 )
=> ( ( image_int_int @ F @ A )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_725_bij__betw__imp__surj__on,axiom,
! [F: set_int > int,A: set_set_int,B2: set_int] :
( ( bij_betw_set_int_int @ F @ A @ B2 )
=> ( ( image_set_int_int @ F @ A )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_726_bij__betw__imp__surj__on,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( ( image_nat_nat @ F @ A )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_727_bij__betw__imp__surj__on,axiom,
! [F: nat > int,A: set_nat,B2: set_int] :
( ( bij_betw_nat_int @ F @ A @ B2 )
=> ( ( image_nat_int @ F @ A )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_728_bij__betw__imp__surj__on,axiom,
! [F: int > set_int,A: set_int,B2: set_set_int] :
( ( bij_betw_int_set_int @ F @ A @ B2 )
=> ( ( image_int_set_int @ F @ A )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_729_bij__betw__imp__surj__on,axiom,
! [F: int > nat,A: set_int,B2: set_nat] :
( ( bij_betw_int_nat @ F @ A @ B2 )
=> ( ( image_int_nat @ F @ A )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_730_bij__betw__imp__surj__on,axiom,
! [F: nat > set_int,A: set_nat,B2: set_set_int] :
( ( bij_betw_nat_set_int @ F @ A @ B2 )
=> ( ( image_nat_set_int @ F @ A )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_731_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_732_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_733_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_734_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_735_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_736_less__add__one,axiom,
! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).
% less_add_one
thf(fact_737_add__mono1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_738_add__mono1,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_739_less__1__mult,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M2 )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_740_less__1__mult,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ one_one_int @ M2 )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_741_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_742_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_743_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_744_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_745_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_746_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_747_range__eq__singletonD,axiom,
! [F: set_int > int,A2: int,X: set_int] :
( ( ( image_set_int_int @ F @ top_top_set_set_int )
= ( insert_int @ A2 @ bot_bot_set_int ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_748_range__eq__singletonD,axiom,
! [F: int > nat,A2: nat,X: int] :
( ( ( image_int_nat @ F @ top_top_set_int )
= ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_749_range__eq__singletonD,axiom,
! [F: int > int,A2: int,X: int] :
( ( ( image_int_int @ F @ top_top_set_int )
= ( insert_int @ A2 @ bot_bot_set_int ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_750_range__eq__singletonD,axiom,
! [F: int > set_int,A2: set_int,X: int] :
( ( ( image_int_set_int @ F @ top_top_set_int )
= ( insert_set_int @ A2 @ bot_bot_set_set_int ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_751_range__eq__singletonD,axiom,
! [F: nat > nat,A2: nat,X: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_752_range__eq__singletonD,axiom,
! [F: nat > int,A2: int,X: nat] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= ( insert_int @ A2 @ bot_bot_set_int ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_753_range__eq__singletonD,axiom,
! [F: nat > set_int,A2: set_int,X: nat] :
( ( ( image_nat_set_int @ F @ top_top_set_nat )
= ( insert_set_int @ A2 @ bot_bot_set_set_int ) )
=> ( ( F @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_754_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_755_pos__zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_756_int_Ogenideal__self_H,axiom,
! [I: int] :
( ( member_int @ I @ top_top_set_int )
=> ( member_int @ I @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ I @ bot_bot_set_int ) ) ) ) ).
% int.genideal_self'
thf(fact_757_int_Ogenideal__zero,axiom,
( ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
= ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ).
% int.genideal_zero
thf(fact_758_int_Ogenideal__one,axiom,
( ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ one_one_int @ bot_bot_set_int ) )
= top_top_set_int ) ).
% int.genideal_one
thf(fact_759_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_760_int_Ol__one,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( times_times_int @ one_one_int @ X )
= X ) ) ).
% int.l_one
thf(fact_761_int_Or__one,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( times_times_int @ X @ one_one_int )
= X ) ) ).
% int.r_one
thf(fact_762_int_Ol__null,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( times_times_int @ zero_zero_int @ X )
= zero_zero_int ) ) ).
% int.l_null
thf(fact_763_UNIV_I3_J,axiom,
! [P: int > $o] :
( ( ! [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
=> ( P @ X3 ) ) )
= ( ! [X7: int] : ( P @ X7 ) ) ) ).
% UNIV(3)
thf(fact_764_UNIV_I3_J,axiom,
! [P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ top_top_set_nat )
=> ( P @ X3 ) ) )
= ( ! [X7: nat] : ( P @ X7 ) ) ) ).
% UNIV(3)
thf(fact_765_UNIV_I4_J,axiom,
! [P: int > $o] :
( ( ? [X3: int] :
( ( member_int @ X3 @ top_top_set_int )
& ( P @ X3 ) ) )
= ( ? [X7: int] : ( P @ X7 ) ) ) ).
% UNIV(4)
thf(fact_766_UNIV_I4_J,axiom,
! [P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ top_top_set_nat )
& ( P @ X3 ) ) )
= ( ? [X7: nat] : ( P @ X7 ) ) ) ).
% UNIV(4)
thf(fact_767_int_Oadd_Oone__closed,axiom,
member_int @ zero_zero_int @ top_top_set_int ).
% int.add.one_closed
thf(fact_768_int_Oadd_Oright__cancel,axiom,
! [X: int,Y: int,Z: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ( ( plus_plus_int @ Y @ X )
= ( plus_plus_int @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% int.add.right_cancel
thf(fact_769_int_Oadd_Om__closed,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( member_int @ ( plus_plus_int @ X @ Y ) @ top_top_set_int ) ) ) ).
% int.add.m_closed
thf(fact_770_int_Oone__closed,axiom,
member_int @ one_one_int @ top_top_set_int ).
% int.one_closed
thf(fact_771_int_Om__closed,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( member_int @ ( times_times_int @ X @ Y ) @ top_top_set_int ) ) ) ).
% int.m_closed
thf(fact_772_int_Oadd_Ol__one,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( plus_plus_int @ zero_zero_int @ X )
= X ) ) ).
% int.add.l_one
thf(fact_773_int_Oadd_Or__one,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( plus_plus_int @ X @ zero_zero_int )
= X ) ) ).
% int.add.r_one
thf(fact_774_int_Oadd_Ol__cancel__one,axiom,
! [X: int,A2: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ A2 @ top_top_set_int )
=> ( ( ( plus_plus_int @ X @ A2 )
= X )
= ( A2 = zero_zero_int ) ) ) ) ).
% int.add.l_cancel_one
thf(fact_775_int_Oadd_Or__cancel__one,axiom,
! [X: int,A2: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ A2 @ top_top_set_int )
=> ( ( ( plus_plus_int @ A2 @ X )
= X )
= ( A2 = zero_zero_int ) ) ) ) ).
% int.add.r_cancel_one
thf(fact_776_int_Oadd_Ol__cancel__one_H,axiom,
! [X: int,A2: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ A2 @ top_top_set_int )
=> ( ( X
= ( plus_plus_int @ X @ A2 ) )
= ( A2 = zero_zero_int ) ) ) ) ).
% int.add.l_cancel_one'
thf(fact_777_int_Oadd_Or__cancel__one_H,axiom,
! [X: int,A2: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ A2 @ top_top_set_int )
=> ( ( X
= ( plus_plus_int @ A2 @ X ) )
= ( A2 = zero_zero_int ) ) ) ) ).
% int.add.r_cancel_one'
thf(fact_778_int_Or__null,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( times_times_int @ X @ zero_zero_int )
= zero_zero_int ) ) ).
% int.r_null
thf(fact_779_top__set__def,axiom,
( top_top_set_int
= ( collect_int @ top_top_int_o ) ) ).
% top_set_def
thf(fact_780_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_781_int_Oadd_Om__lcomm,axiom,
! [X: int,Y: int,Z: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ( plus_plus_int @ X @ ( plus_plus_int @ Y @ Z ) )
= ( plus_plus_int @ Y @ ( plus_plus_int @ X @ Z ) ) ) ) ) ) ).
% int.add.m_lcomm
thf(fact_782_int_Oadd_Om__assoc,axiom,
! [X: int,Y: int,Z: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ( plus_plus_int @ ( plus_plus_int @ X @ Y ) @ Z )
= ( plus_plus_int @ X @ ( plus_plus_int @ Y @ Z ) ) ) ) ) ) ).
% int.add.m_assoc
thf(fact_783_int_Oadd_Om__comm,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( plus_plus_int @ X @ Y )
= ( plus_plus_int @ Y @ X ) ) ) ) ).
% int.add.m_comm
thf(fact_784_int_Om__lcomm,axiom,
! [X: int,Y: int,Z: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ( times_times_int @ X @ ( times_times_int @ Y @ Z ) )
= ( times_times_int @ Y @ ( times_times_int @ X @ Z ) ) ) ) ) ) ).
% int.m_lcomm
thf(fact_785_int_Om__assoc,axiom,
! [X: int,Y: int,Z: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ( times_times_int @ ( times_times_int @ X @ Y ) @ Z )
= ( times_times_int @ X @ ( times_times_int @ Y @ Z ) ) ) ) ) ) ).
% int.m_assoc
thf(fact_786_int_Om__comm,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) ) ) ) ).
% int.m_comm
thf(fact_787_int_Ozero__not__one,axiom,
zero_zero_int != one_one_int ).
% int.zero_not_one
thf(fact_788_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_789_int_Olless__antisym,axiom,
! [A2: int,B: int] :
( ( member_int @ A2 @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( ord_less_int @ A2 @ B )
=> ~ ( ord_less_int @ B @ A2 ) ) ) ) ).
% int.lless_antisym
thf(fact_790_int_Olless__trans,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ( member_int @ A2 @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C2 @ top_top_set_int )
=> ( ord_less_int @ A2 @ C2 ) ) ) ) ) ) ).
% int.lless_trans
thf(fact_791_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_792_int_Oadd_Ocarrier__not__empty,axiom,
top_top_set_int != bot_bot_set_int ).
% int.add.carrier_not_empty
thf(fact_793_int_Oadd_Oinv__comm,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( plus_plus_int @ Y @ X )
= zero_zero_int ) ) ) ) ).
% int.add.inv_comm
thf(fact_794_int_Oadd_Ol__inv__ex,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ top_top_set_int )
& ( ( plus_plus_int @ X2 @ X )
= zero_zero_int ) ) ) ).
% int.add.l_inv_ex
thf(fact_795_int_Oadd_Or__inv__ex,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ? [X2: int] :
( ( member_int @ X2 @ top_top_set_int )
& ( ( plus_plus_int @ X @ X2 )
= zero_zero_int ) ) ) ).
% int.add.r_inv_ex
thf(fact_796_int_Oadd_Oinv__unique,axiom,
! [Y: int,X: int,Y6: int] :
( ( ( plus_plus_int @ Y @ X )
= zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y6 )
= zero_zero_int )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Y6 @ top_top_set_int )
=> ( Y = Y6 ) ) ) ) ) ) ).
% int.add.inv_unique
thf(fact_797_int_Oadd_Oone__unique,axiom,
! [U: int] :
( ( member_int @ U @ top_top_set_int )
=> ( ! [X2: int] :
( ( member_int @ X2 @ top_top_set_int )
=> ( ( plus_plus_int @ U @ X2 )
= X2 ) )
=> ( U = zero_zero_int ) ) ) ).
% int.add.one_unique
thf(fact_798_int_Ointegral__iff,axiom,
! [A2: int,B: int] :
( ( member_int @ A2 @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ) ) ).
% int.integral_iff
thf(fact_799_int_Om__rcancel,axiom,
! [A2: int,B: int,C2: int] :
( ( A2 != zero_zero_int )
=> ( ( member_int @ A2 @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C2 @ top_top_set_int )
=> ( ( ( times_times_int @ B @ A2 )
= ( times_times_int @ C2 @ A2 ) )
= ( B = C2 ) ) ) ) ) ) ).
% int.m_rcancel
thf(fact_800_int_Om__lcancel,axiom,
! [A2: int,B: int,C2: int] :
( ( A2 != zero_zero_int )
=> ( ( member_int @ A2 @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C2 @ top_top_set_int )
=> ( ( ( times_times_int @ A2 @ B )
= ( times_times_int @ A2 @ C2 ) )
= ( B = C2 ) ) ) ) ) ) ).
% int.m_lcancel
thf(fact_801_int_Ointegral,axiom,
! [A2: int,B: int] :
( ( ( times_times_int @ A2 @ B )
= zero_zero_int )
=> ( ( member_int @ A2 @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( A2 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ) ) ).
% int.integral
thf(fact_802_int_Or__distr,axiom,
! [X: int,Y: int,Z: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ( times_times_int @ Z @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) ) ) ) ) ) ).
% int.r_distr
thf(fact_803_int_Ol__distr,axiom,
! [X: int,Y: int,Z: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ( times_times_int @ ( plus_plus_int @ X @ Y ) @ Z )
= ( plus_plus_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) ) ) ) ) ) ).
% int.l_distr
thf(fact_804_int_Oone__unique,axiom,
! [U: int] :
( ( member_int @ U @ top_top_set_int )
=> ( ! [X2: int] :
( ( member_int @ X2 @ top_top_set_int )
=> ( ( times_times_int @ U @ X2 )
= X2 ) )
=> ( U = one_one_int ) ) ) ).
% int.one_unique
thf(fact_805_int_Oinv__unique,axiom,
! [Y: int,X: int,Y6: int] :
( ( ( times_times_int @ Y @ X )
= one_one_int )
=> ( ( ( times_times_int @ X @ Y6 )
= one_one_int )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Y6 @ top_top_set_int )
=> ( Y = Y6 ) ) ) ) ) ) ).
% int.inv_unique
thf(fact_806_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_807_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_808_ZFact__one,axiom,
( ( partia966996272515721803t_unit @ ( zFact @ one_one_int ) )
= ( insert_set_int @ top_top_set_int @ bot_bot_set_set_int ) ) ).
% ZFact_one
thf(fact_809_int_Oone__zeroD,axiom,
( ( one_one_int = zero_zero_int )
=> ( top_top_set_int
= ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ) ).
% int.one_zeroD
thf(fact_810_int_Oone__zeroI,axiom,
( ( top_top_set_int
= ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
=> ( one_one_int = zero_zero_int ) ) ).
% int.one_zeroI
thf(fact_811_int_Ocarrier__one__zero,axiom,
( ( top_top_set_int
= ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
= ( one_one_int = zero_zero_int ) ) ).
% int.carrier_one_zero
thf(fact_812_int_Ocarrier__one__not__zero,axiom,
( ( top_top_set_int
!= ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
= ( one_one_int != zero_zero_int ) ) ).
% int.carrier_one_not_zero
thf(fact_813_range__add,axiom,
! [A2: int] :
( ( image_int_int @ ( plus_plus_int @ A2 ) @ top_top_set_int )
= top_top_set_int ) ).
% range_add
thf(fact_814_int_Ozeropideal,axiom,
princi1768892856804252751t_unit @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ).
% int.zeropideal
thf(fact_815_ZFact__def,axiom,
( zFact
= ( ^ [K3: int] : ( factRi5755170488246124606t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ K3 @ bot_bot_set_int ) ) ) ) ) ).
% ZFact_def
thf(fact_816_ring_Ocases,axiom,
! [R: partia2818514838349642498t_unit] :
~ ! [Carrier: set_int,Mult: int > int > int,One: int,Zero: int,Add: int > int > int] :
( R
!= ( partia4118392927963588428t_unit @ Carrier @ ( monoid6080699973261426200t_unit @ Mult @ One @ ( ring_e5272872978682396362t_unit @ Zero @ Add @ product_Unity ) ) ) ) ).
% ring.cases
thf(fact_817_zfact__iso__def,axiom,
( ring_zfact_iso
= ( ^ [P4: nat,K3: nat] : ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ ( semiri1314217659103216013at_int @ P4 ) @ bot_bot_set_int ) ) @ ( semiri1314217659103216013at_int @ K3 ) ) ) ) ).
% zfact_iso_def
thf(fact_818_ring_Oext__inject,axiom,
! [Zero2: int,Add2: int > int > int,More: product_unit,Zero3: int,Add3: int > int > int,More2: product_unit] :
( ( ( ring_e5272872978682396362t_unit @ Zero2 @ Add2 @ More )
= ( ring_e5272872978682396362t_unit @ Zero3 @ Add3 @ More2 ) )
= ( ( Zero2 = Zero3 )
& ( Add2 = Add3 )
& ( More = More2 ) ) ) ).
% ring.ext_inject
thf(fact_819_ring_Oext__induct,axiom,
! [P: ring_e6626950497611839816t_unit > $o,R: ring_e6626950497611839816t_unit] :
( ! [Zero: int,Add: int > int > int,More3: product_unit] : ( P @ ( ring_e5272872978682396362t_unit @ Zero @ Add @ More3 ) )
=> ( P @ R ) ) ).
% ring.ext_induct
thf(fact_820_ring_Ocases__scheme,axiom,
! [R: partia2818514838349642498t_unit] :
~ ! [Carrier: set_int,Mult: int > int > int,One: int,Zero: int,Add: int > int > int,More3: product_unit] :
( R
!= ( partia4118392927963588428t_unit @ Carrier @ ( monoid6080699973261426200t_unit @ Mult @ One @ ( ring_e5272872978682396362t_unit @ Zero @ Add @ More3 ) ) ) ) ).
% ring.cases_scheme
thf(fact_821_int_Oonepideal,axiom,
princi1768892856804252751t_unit @ top_top_set_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ).
% int.onepideal
thf(fact_822_ZMod__def,axiom,
( zMod
= ( ^ [K3: int] : ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ K3 @ bot_bot_set_int ) ) ) ) ) ).
% ZMod_def
thf(fact_823_int_OFactRing__zeroideal_I2_J,axiom,
is_rin1886641436590440976t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( factRi5755170488246124606t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ).
% int.FactRing_zeroideal(2)
thf(fact_824_int_OFactRing__zeroideal_I1_J,axiom,
is_rin6476721666283997948t_unit @ ( factRi5755170488246124606t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ).
% int.FactRing_zeroideal(1)
thf(fact_825_principalideal_Ogenerate,axiom,
! [I3: set_set_int,R2: partia4934656038542163276t_unit] :
( ( princi8860937869964495385t_unit @ I3 @ R2 )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ ( partia966996272515721803t_unit @ R2 ) )
& ( I3
= ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ X2 @ bot_bot_set_set_int ) ) ) ) ) ).
% principalideal.generate
thf(fact_826_principalideal_Ogenerate,axiom,
! [I3: set_int,R2: partia2818514838349642498t_unit] :
( ( princi1768892856804252751t_unit @ I3 @ R2 )
=> ? [X2: int] :
( ( member_int @ X2 @ ( partia8426541738980984321t_unit @ R2 ) )
& ( I3
= ( genide1613390280493775889t_unit @ R2 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) ).
% principalideal.generate
thf(fact_827_int__cosetI,axiom,
! [N: nat,U: int,V: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ( modulo_modulo_int @ U @ ( semiri1314217659103216013at_int @ N ) )
= ( modulo_modulo_int @ V @ ( semiri1314217659103216013at_int @ N ) ) )
=> ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ ( semiri1314217659103216013at_int @ N ) @ bot_bot_set_int ) ) @ U )
= ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ ( semiri1314217659103216013at_int @ N ) @ bot_bot_set_int ) ) @ V ) ) ) ) ).
% int_cosetI
thf(fact_828_mod__self,axiom,
! [A2: int] :
( ( modulo_modulo_int @ A2 @ A2 )
= zero_zero_int ) ).
% mod_self
thf(fact_829_mod__self,axiom,
! [A2: nat] :
( ( modulo_modulo_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% mod_self
thf(fact_830_mod__by__0,axiom,
! [A2: int] :
( ( modulo_modulo_int @ A2 @ zero_zero_int )
= A2 ) ).
% mod_by_0
thf(fact_831_mod__by__0,axiom,
! [A2: nat] :
( ( modulo_modulo_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% mod_by_0
thf(fact_832_mod__0,axiom,
! [A2: int] :
( ( modulo_modulo_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% mod_0
thf(fact_833_mod__0,axiom,
! [A2: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mod_0
thf(fact_834_mod__by__1,axiom,
! [A2: int] :
( ( modulo_modulo_int @ A2 @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_835_mod__by__1,axiom,
! [A2: nat] :
( ( modulo_modulo_nat @ A2 @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_836_int__Zcarr,axiom,
! [K: int] : ( member_int @ K @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ).
% int_Zcarr
thf(fact_837_rcos__zfact,axiom,
! [K: int,L: int,R: int] :
( ( member_int @ K @ ( zMod @ L @ R ) )
=> ? [X2: int] :
( K
= ( plus_plus_int @ ( times_times_int @ X2 @ L ) @ R ) ) ) ).
% rcos_zfact
thf(fact_838_int__carrier__eq,axiom,
( ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) )
= top_top_set_int ) ).
% int_carrier_eq
thf(fact_839_mod__mult__self1,axiom,
! [A2: int,C2: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A2 @ ( times_times_int @ C2 @ B ) ) @ B )
= ( modulo_modulo_int @ A2 @ B ) ) ).
% mod_mult_self1
thf(fact_840_mod__mult__self1,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ C2 @ B ) ) @ B )
= ( modulo_modulo_nat @ A2 @ B ) ) ).
% mod_mult_self1
thf(fact_841_mod__mult__self2,axiom,
! [A2: int,B: int,C2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A2 @ ( times_times_int @ B @ C2 ) ) @ B )
= ( modulo_modulo_int @ A2 @ B ) ) ).
% mod_mult_self2
thf(fact_842_mod__mult__self2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ B @ C2 ) ) @ B )
= ( modulo_modulo_nat @ A2 @ B ) ) ).
% mod_mult_self2
thf(fact_843_mod__mult__self3,axiom,
! [C2: int,B: int,A2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C2 @ B ) @ A2 ) @ B )
= ( modulo_modulo_int @ A2 @ B ) ) ).
% mod_mult_self3
thf(fact_844_mod__mult__self3,axiom,
! [C2: nat,B: nat,A2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B ) @ A2 ) @ B )
= ( modulo_modulo_nat @ A2 @ B ) ) ).
% mod_mult_self3
thf(fact_845_mod__mult__self4,axiom,
! [B: int,C2: int,A2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C2 ) @ A2 ) @ B )
= ( modulo_modulo_int @ A2 @ B ) ) ).
% mod_mult_self4
thf(fact_846_mod__mult__self4,axiom,
! [B: nat,C2: nat,A2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C2 ) @ A2 ) @ B )
= ( modulo_modulo_nat @ A2 @ B ) ) ).
% mod_mult_self4
thf(fact_847_bits__mod__by__1,axiom,
! [A2: int] :
( ( modulo_modulo_int @ A2 @ one_one_int )
= zero_zero_int ) ).
% bits_mod_by_1
thf(fact_848_bits__mod__by__1,axiom,
! [A2: nat] :
( ( modulo_modulo_nat @ A2 @ one_one_nat )
= zero_zero_nat ) ).
% bits_mod_by_1
thf(fact_849_mod__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( modulo_modulo_nat @ M2 @ N )
= M2 ) ) ).
% mod_less
thf(fact_850_bits__mod__0,axiom,
! [A2: int] :
( ( modulo_modulo_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% bits_mod_0
thf(fact_851_bits__mod__0,axiom,
! [A2: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% bits_mod_0
thf(fact_852_mod__add__self2,axiom,
! [A2: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( modulo_modulo_int @ A2 @ B ) ) ).
% mod_add_self2
thf(fact_853_mod__add__self2,axiom,
! [A2: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( modulo_modulo_nat @ A2 @ B ) ) ).
% mod_add_self2
thf(fact_854_mod__add__self1,axiom,
! [B: int,A2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( modulo_modulo_int @ A2 @ B ) ) ).
% mod_add_self1
thf(fact_855_mod__add__self1,axiom,
! [B: nat,A2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( modulo_modulo_nat @ A2 @ B ) ) ).
% mod_add_self1
thf(fact_856_mod__mult__self2__is__0,axiom,
! [A2: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A2 @ B ) @ B )
= zero_zero_int ) ).
% mod_mult_self2_is_0
thf(fact_857_mod__mult__self2__is__0,axiom,
! [A2: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B ) @ B )
= zero_zero_nat ) ).
% mod_mult_self2_is_0
thf(fact_858_mod__mult__self1__is__0,axiom,
! [B: int,A2: int] :
( ( modulo_modulo_int @ ( times_times_int @ B @ A2 ) @ B )
= zero_zero_int ) ).
% mod_mult_self1_is_0
thf(fact_859_mod__mult__self1__is__0,axiom,
! [B: nat,A2: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ B @ A2 ) @ B )
= zero_zero_nat ) ).
% mod_mult_self1_is_0
thf(fact_860_mod__less__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_861_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo_nat @ X @ N )
= ( modulo_modulo_nat @ Y @ N ) )
= ( ? [Q1: nat,Q2: nat] :
( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
= ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q2 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_862_zmod__int,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M2 @ N ) )
= ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% zmod_int
thf(fact_863_int__ops_I9_J,axiom,
! [A2: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A2 @ B ) )
= ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(9)
thf(fact_864_split__mod,axiom,
! [Q: nat > $o,M2: nat,N: nat] :
( ( Q @ ( modulo_modulo_nat @ M2 @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( Q @ M2 ) )
& ( ( N != zero_zero_nat )
=> ! [I4: nat,J2: nat] :
( ( ( ord_less_nat @ J2 @ N )
& ( M2
= ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J2 ) ) )
=> ( Q @ J2 ) ) ) ) ) ).
% split_mod
thf(fact_865_mod__add__eq,axiom,
! [A2: int,C2: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A2 @ C2 ) @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 )
= ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B ) @ C2 ) ) ).
% mod_add_eq
thf(fact_866_mod__add__eq,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ C2 ) @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 ) ) ).
% mod_add_eq
thf(fact_867_mod__add__cong,axiom,
! [A2: int,C2: int,A7: int,B: int,B7: int] :
( ( ( modulo_modulo_int @ A2 @ C2 )
= ( modulo_modulo_int @ A7 @ C2 ) )
=> ( ( ( modulo_modulo_int @ B @ C2 )
= ( modulo_modulo_int @ B7 @ C2 ) )
=> ( ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B ) @ C2 )
= ( modulo_modulo_int @ ( plus_plus_int @ A7 @ B7 ) @ C2 ) ) ) ) ).
% mod_add_cong
thf(fact_868_mod__add__cong,axiom,
! [A2: nat,C2: nat,A7: nat,B: nat,B7: nat] :
( ( ( modulo_modulo_nat @ A2 @ C2 )
= ( modulo_modulo_nat @ A7 @ C2 ) )
=> ( ( ( modulo_modulo_nat @ B @ C2 )
= ( modulo_modulo_nat @ B7 @ C2 ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A7 @ B7 ) @ C2 ) ) ) ) ).
% mod_add_cong
thf(fact_869_mod__add__left__eq,axiom,
! [A2: int,C2: int,B: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A2 @ C2 ) @ B ) @ C2 )
= ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B ) @ C2 ) ) ).
% mod_add_left_eq
thf(fact_870_mod__add__left__eq,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ C2 ) @ B ) @ C2 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 ) ) ).
% mod_add_left_eq
thf(fact_871_mod__add__right__eq,axiom,
! [A2: int,B: int,C2: int] :
( ( modulo_modulo_int @ ( plus_plus_int @ A2 @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 )
= ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B ) @ C2 ) ) ).
% mod_add_right_eq
thf(fact_872_mod__add__right__eq,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B ) @ C2 ) ) ).
% mod_add_right_eq
thf(fact_873_mod__mult__eq,axiom,
! [A2: int,C2: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A2 @ C2 ) @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 )
= ( modulo_modulo_int @ ( times_times_int @ A2 @ B ) @ C2 ) ) ).
% mod_mult_eq
thf(fact_874_mod__mult__eq,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A2 @ C2 ) @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 )
= ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B ) @ C2 ) ) ).
% mod_mult_eq
thf(fact_875_mod__mult__cong,axiom,
! [A2: int,C2: int,A7: int,B: int,B7: int] :
( ( ( modulo_modulo_int @ A2 @ C2 )
= ( modulo_modulo_int @ A7 @ C2 ) )
=> ( ( ( modulo_modulo_int @ B @ C2 )
= ( modulo_modulo_int @ B7 @ C2 ) )
=> ( ( modulo_modulo_int @ ( times_times_int @ A2 @ B ) @ C2 )
= ( modulo_modulo_int @ ( times_times_int @ A7 @ B7 ) @ C2 ) ) ) ) ).
% mod_mult_cong
thf(fact_876_mod__mult__cong,axiom,
! [A2: nat,C2: nat,A7: nat,B: nat,B7: nat] :
( ( ( modulo_modulo_nat @ A2 @ C2 )
= ( modulo_modulo_nat @ A7 @ C2 ) )
=> ( ( ( modulo_modulo_nat @ B @ C2 )
= ( modulo_modulo_nat @ B7 @ C2 ) )
=> ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B ) @ C2 )
= ( modulo_modulo_nat @ ( times_times_nat @ A7 @ B7 ) @ C2 ) ) ) ) ).
% mod_mult_cong
thf(fact_877_mod__mult__mult2,axiom,
! [A2: int,C2: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B @ C2 ) )
= ( times_times_int @ ( modulo_modulo_int @ A2 @ B ) @ C2 ) ) ).
% mod_mult_mult2
thf(fact_878_mod__mult__mult2,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B @ C2 ) )
= ( times_times_nat @ ( modulo_modulo_nat @ A2 @ B ) @ C2 ) ) ).
% mod_mult_mult2
thf(fact_879_mult__mod__right,axiom,
! [C2: int,A2: int,B: int] :
( ( times_times_int @ C2 @ ( modulo_modulo_int @ A2 @ B ) )
= ( modulo_modulo_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B ) ) ) ).
% mult_mod_right
thf(fact_880_mult__mod__right,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( times_times_nat @ C2 @ ( modulo_modulo_nat @ A2 @ B ) )
= ( modulo_modulo_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B ) ) ) ).
% mult_mod_right
thf(fact_881_mod__mult__left__eq,axiom,
! [A2: int,C2: int,B: int] :
( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A2 @ C2 ) @ B ) @ C2 )
= ( modulo_modulo_int @ ( times_times_int @ A2 @ B ) @ C2 ) ) ).
% mod_mult_left_eq
thf(fact_882_mod__mult__left__eq,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A2 @ C2 ) @ B ) @ C2 )
= ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B ) @ C2 ) ) ).
% mod_mult_left_eq
thf(fact_883_mod__mult__right__eq,axiom,
! [A2: int,B: int,C2: int] :
( ( modulo_modulo_int @ ( times_times_int @ A2 @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 )
= ( modulo_modulo_int @ ( times_times_int @ A2 @ B ) @ C2 ) ) ).
% mod_mult_right_eq
thf(fact_884_mod__mult__right__eq,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 )
= ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B ) @ C2 ) ) ).
% mod_mult_right_eq
thf(fact_885_mod__eqE,axiom,
! [A2: int,C2: int,B: int] :
( ( ( modulo_modulo_int @ A2 @ C2 )
= ( modulo_modulo_int @ B @ C2 ) )
=> ~ ! [D3: int] :
( B
!= ( plus_plus_int @ A2 @ ( times_times_int @ C2 @ D3 ) ) ) ) ).
% mod_eqE
thf(fact_886_neg__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% neg_mod_bound
thf(fact_887_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_888_int_Ozeroprimeideal,axiom,
primei2109666362732673920t_unit @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ).
% int.zeroprimeideal
thf(fact_889_int_Ocgenideal__eq__genideal,axiom,
! [I: int] :
( ( member_int @ I @ top_top_set_int )
=> ( ( cgenid7570434961818237563t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I )
= ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ I @ bot_bot_set_int ) ) ) ) ).
% int.cgenideal_eq_genideal
thf(fact_890_int_Oa__coset__add__zero,axiom,
! [M: set_int] :
( ( ord_less_eq_set_int @ M @ top_top_set_int )
=> ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ zero_zero_int )
= M ) ) ).
% int.a_coset_add_zero
thf(fact_891_int__Idl__subset__ideal,axiom,
! [K: int,L: int] :
( ( ord_less_eq_set_int @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ K @ bot_bot_set_int ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ L @ bot_bot_set_int ) ) )
= ( member_int @ K @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ L @ bot_bot_set_int ) ) ) ) ).
% int_Idl_subset_ideal
thf(fact_892_order__refl,axiom,
! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).
% order_refl
thf(fact_893_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_894_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_895_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_896_dual__order_Orefl,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_897_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_898_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_899_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_900_subsetI,axiom,
! [A: set_set_int,B2: set_set_int] :
( ! [X2: set_int] :
( ( member_set_int @ X2 @ A )
=> ( member_set_int @ X2 @ B2 ) )
=> ( ord_le4403425263959731960et_int @ A @ B2 ) ) ).
% subsetI
thf(fact_901_subsetI,axiom,
! [A: set_int,B2: set_int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_int @ X2 @ B2 ) )
=> ( ord_less_eq_set_int @ A @ B2 ) ) ).
% subsetI
thf(fact_902_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_903_subset__antisym,axiom,
! [A: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_904_subset__antisym,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_905_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_906_add__le__cancel__right,axiom,
! [A2: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_907_add__le__cancel__right,axiom,
! [A2: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_908_add__le__cancel__left,axiom,
! [C2: int,A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_eq_int @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_909_add__le__cancel__left,axiom,
! [C2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_910_subset__empty,axiom,
! [A: set_set_int] :
( ( ord_le4403425263959731960et_int @ A @ bot_bot_set_set_int )
= ( A = bot_bot_set_set_int ) ) ).
% subset_empty
thf(fact_911_subset__empty,axiom,
! [A: set_int] :
( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
= ( A = bot_bot_set_int ) ) ).
% subset_empty
thf(fact_912_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_913_empty__subsetI,axiom,
! [A: set_set_int] : ( ord_le4403425263959731960et_int @ bot_bot_set_set_int @ A ) ).
% empty_subsetI
thf(fact_914_empty__subsetI,axiom,
! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% empty_subsetI
thf(fact_915_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_916_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_917_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_918_insert__subset,axiom,
! [X: set_int,A: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( insert_set_int @ X @ A ) @ B2 )
= ( ( member_set_int @ X @ B2 )
& ( ord_le4403425263959731960et_int @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_919_insert__subset,axiom,
! [X: int,A: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ ( insert_int @ X @ A ) @ B2 )
= ( ( member_int @ X @ B2 )
& ( ord_less_eq_set_int @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_920_insert__subset,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B2 )
= ( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_921_psubsetI,axiom,
! [A: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_int @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_922_psubsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_nat @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_923_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_924_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_925_le__add__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_926_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_927_le__add__same__cancel1,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_928_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_929_add__le__same__cancel2,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B ) @ B )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_930_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_931_add__le__same__cancel1,axiom,
! [B: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A2 ) @ B )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_932_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_933_singleton__insert__inj__eq,axiom,
! [B: set_int,A2: set_int,A: set_set_int] :
( ( ( insert_set_int @ B @ bot_bot_set_set_int )
= ( insert_set_int @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le4403425263959731960et_int @ A @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_934_singleton__insert__inj__eq,axiom,
! [B: int,A2: int,A: set_int] :
( ( ( insert_int @ B @ bot_bot_set_int )
= ( insert_int @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_int @ A @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_935_singleton__insert__inj__eq,axiom,
! [B: nat,A2: nat,A: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_936_singleton__insert__inj__eq_H,axiom,
! [A2: set_int,A: set_set_int,B: set_int] :
( ( ( insert_set_int @ A2 @ A )
= ( insert_set_int @ B @ bot_bot_set_set_int ) )
= ( ( A2 = B )
& ( ord_le4403425263959731960et_int @ A @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_937_singleton__insert__inj__eq_H,axiom,
! [A2: int,A: set_int,B: int] :
( ( ( insert_int @ A2 @ A )
= ( insert_int @ B @ bot_bot_set_int ) )
= ( ( A2 = B )
& ( ord_less_eq_set_int @ A @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_938_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B: nat] :
( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A2 = B )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_939_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_940_lessThan__subset__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_941_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_942_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_943_subset__insertI2,axiom,
! [A: set_set_int,B2: set_set_int,B: set_int] :
( ( ord_le4403425263959731960et_int @ A @ B2 )
=> ( ord_le4403425263959731960et_int @ A @ ( insert_set_int @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_944_subset__insertI2,axiom,
! [A: set_int,B2: set_int,B: int] :
( ( ord_less_eq_set_int @ A @ B2 )
=> ( ord_less_eq_set_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_945_subset__insertI2,axiom,
! [A: set_nat,B2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_946_subset__insertI,axiom,
! [B2: set_set_int,A2: set_int] : ( ord_le4403425263959731960et_int @ B2 @ ( insert_set_int @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_947_subset__insertI,axiom,
! [B2: set_int,A2: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_948_subset__insertI,axiom,
! [B2: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_949_subset__insert,axiom,
! [X: set_int,A: set_set_int,B2: set_set_int] :
( ~ ( member_set_int @ X @ A )
=> ( ( ord_le4403425263959731960et_int @ A @ ( insert_set_int @ X @ B2 ) )
= ( ord_le4403425263959731960et_int @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_950_subset__insert,axiom,
! [X: int,A: set_int,B2: set_int] :
( ~ ( member_int @ X @ A )
=> ( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B2 ) )
= ( ord_less_eq_set_int @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_951_subset__insert,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_952_insert__mono,axiom,
! [C: set_set_int,D: set_set_int,A2: set_int] :
( ( ord_le4403425263959731960et_int @ C @ D )
=> ( ord_le4403425263959731960et_int @ ( insert_set_int @ A2 @ C ) @ ( insert_set_int @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_953_insert__mono,axiom,
! [C: set_int,D: set_int,A2: int] :
( ( ord_less_eq_set_int @ C @ D )
=> ( ord_less_eq_set_int @ ( insert_int @ A2 @ C ) @ ( insert_int @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_954_insert__mono,axiom,
! [C: set_nat,D: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C ) @ ( insert_nat @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_955_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_956_verit__comp__simplify1_I3_J,axiom,
! [B7: int,A7: int] :
( ( ~ ( ord_less_eq_int @ B7 @ A7 ) )
= ( ord_less_int @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_957_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
= ( ord_less_nat @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_958_leD,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y @ X )
=> ~ ( ord_less_set_int @ X @ Y ) ) ).
% leD
thf(fact_959_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_960_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_961_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_962_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_963_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_964_nless__le,axiom,
! [A2: set_int,B: set_int] :
( ( ~ ( ord_less_set_int @ A2 @ B ) )
= ( ~ ( ord_less_eq_set_int @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_965_nless__le,axiom,
! [A2: int,B: int] :
( ( ~ ( ord_less_int @ A2 @ B ) )
= ( ~ ( ord_less_eq_int @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_966_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_967_nless__le,axiom,
! [A2: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_968_antisym__conv1,axiom,
! [X: set_int,Y: set_int] :
( ~ ( ord_less_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_969_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_970_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_971_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_972_antisym__conv2,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ~ ( ord_less_set_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_973_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_974_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_975_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_976_less__le__not__le,axiom,
( ord_less_set_int
= ( ^ [X3: set_int,Y5: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y5 )
& ~ ( ord_less_eq_set_int @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_977_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y5: int] :
( ( ord_less_eq_int @ X3 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_978_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_979_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
& ~ ( ord_less_eq_set_nat @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_980_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_981_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_982_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_int
= ( ^ [A3: set_int,B3: set_int] :
( ( ord_less_set_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_983_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_984_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_985_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_986_order_Ostrict__iff__order,axiom,
( ord_less_set_int
= ( ^ [A3: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_987_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_988_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_989_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_990_order_Ostrict__trans1,axiom,
! [A2: set_int,B: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ A2 @ B )
=> ( ( ord_less_set_int @ B @ C2 )
=> ( ord_less_set_int @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_991_order_Ostrict__trans1,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_992_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_993_order_Ostrict__trans1,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_994_order_Ostrict__trans2,axiom,
! [A2: set_int,B: set_int,C2: set_int] :
( ( ord_less_set_int @ A2 @ B )
=> ( ( ord_less_eq_set_int @ B @ C2 )
=> ( ord_less_set_int @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_995_order_Ostrict__trans2,axiom,
! [A2: int,B: int,C2: int] :
( ( ord_less_int @ A2 @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_996_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_997_order_Ostrict__trans2,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_998_order_Ostrict__iff__not,axiom,
( ord_less_set_int
= ( ^ [A3: set_int,B3: set_int] :
( ( ord_less_eq_set_int @ A3 @ B3 )
& ~ ( ord_less_eq_set_int @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_999_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1000_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1001_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1002_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_int
= ( ^ [B3: set_int,A3: set_int] :
( ( ord_less_set_int @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1003_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1004_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1005_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( ord_less_set_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1006_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_int
= ( ^ [B3: set_int,A3: set_int] :
( ( ord_less_eq_set_int @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1007_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1008_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1009_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1010_dual__order_Ostrict__trans1,axiom,
! [B: set_int,A2: set_int,C2: set_int] :
( ( ord_less_eq_set_int @ B @ A2 )
=> ( ( ord_less_set_int @ C2 @ B )
=> ( ord_less_set_int @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1011_dual__order_Ostrict__trans1,axiom,
! [B: int,A2: int,C2: int] :
( ( ord_less_eq_int @ B @ A2 )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1012_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1013_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_set_nat @ C2 @ B )
=> ( ord_less_set_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1014_dual__order_Ostrict__trans2,axiom,
! [B: set_int,A2: set_int,C2: set_int] :
( ( ord_less_set_int @ B @ A2 )
=> ( ( ord_less_eq_set_int @ C2 @ B )
=> ( ord_less_set_int @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1015_dual__order_Ostrict__trans2,axiom,
! [B: int,A2: int,C2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1016_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1017_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A2: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_set_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1018_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_int
= ( ^ [B3: set_int,A3: set_int] :
( ( ord_less_eq_set_int @ B3 @ A3 )
& ~ ( ord_less_eq_set_int @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1019_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1020_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1021_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ~ ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1022_order_Ostrict__implies__order,axiom,
! [A2: set_int,B: set_int] :
( ( ord_less_set_int @ A2 @ B )
=> ( ord_less_eq_set_int @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1023_order_Ostrict__implies__order,axiom,
! [A2: int,B: int] :
( ( ord_less_int @ A2 @ B )
=> ( ord_less_eq_int @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1024_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1025_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_1026_dual__order_Ostrict__implies__order,axiom,
! [B: set_int,A2: set_int] :
( ( ord_less_set_int @ B @ A2 )
=> ( ord_less_eq_set_int @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1027_dual__order_Ostrict__implies__order,axiom,
! [B: int,A2: int] :
( ( ord_less_int @ B @ A2 )
=> ( ord_less_eq_int @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1028_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1029_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1030_order__le__less,axiom,
( ord_less_eq_set_int
= ( ^ [X3: set_int,Y5: set_int] :
( ( ord_less_set_int @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1031_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y5: int] :
( ( ord_less_int @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1032_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1033_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_1034_order__less__le,axiom,
( ord_less_set_int
= ( ^ [X3: set_int,Y5: set_int] :
( ( ord_less_eq_set_int @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1035_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y5: int] :
( ( ord_less_eq_int @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1036_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1037_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_1038_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1039_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1040_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1041_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1042_order__less__imp__le,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_set_int @ X @ Y )
=> ( ord_less_eq_set_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1043_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1044_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1045_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1046_order__le__neq__trans,axiom,
! [A2: set_int,B: set_int] :
( ( ord_less_eq_set_int @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_int @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1047_order__le__neq__trans,axiom,
! [A2: int,B: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1048_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1049_order__le__neq__trans,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1050_order__neq__le__trans,axiom,
! [A2: set_int,B: set_int] :
( ( A2 != B )
=> ( ( ord_less_eq_set_int @ A2 @ B )
=> ( ord_less_set_int @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1051_order__neq__le__trans,axiom,
! [A2: int,B: int] :
( ( A2 != B )
=> ( ( ord_less_eq_int @ A2 @ B )
=> ( ord_less_int @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1052_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1053_order__neq__le__trans,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 != B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1054_order__le__less__trans,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ( ord_less_set_int @ Y @ Z )
=> ( ord_less_set_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1055_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1056_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1057_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_1058_order__less__le__trans,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( ord_less_set_int @ X @ Y )
=> ( ( ord_less_eq_set_int @ Y @ Z )
=> ( ord_less_set_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1059_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1060_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1061_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_1062_order__le__less__subst1,axiom,
! [A2: set_int,F: nat > set_int,B: nat,C2: nat] :
( ( ord_less_eq_set_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1063_order__le__less__subst1,axiom,
! [A2: set_int,F: int > set_int,B: int,C2: int] :
( ( ord_less_eq_set_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_set_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1064_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1065_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1066_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1067_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1068_order__le__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1069_order__le__less__subst1,axiom,
! [A2: set_nat,F: int > set_nat,B: int,C2: int] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1070_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1071_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1072_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1073_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1074_order__le__less__subst2,axiom,
! [A2: set_int,B: set_int,F: set_int > int,C2: int] :
( ( ord_less_eq_set_int @ A2 @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X2: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1075_order__le__less__subst2,axiom,
! [A2: set_int,B: set_int,F: set_int > nat,C2: nat] :
( ( ord_less_eq_set_int @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: set_int,Y2: set_int] :
( ( ord_less_eq_set_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1076_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > set_int,C2: set_int] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_set_int @ ( F @ B ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_set_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1077_order__le__less__subst2,axiom,
! [A2: int,B: int,F: int > set_nat,C2: set_nat] :
( ( ord_less_eq_int @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1078_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_int,C2: set_int] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_set_int @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_int @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1079_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1080_order__less__le__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C2: set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1081_order__less__le__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1082_int_Ocgenideal__self,axiom,
! [I: int] :
( ( member_int @ I @ top_top_set_int )
=> ( member_int @ I @ ( cgenid7570434961818237563t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I ) ) ) ).
% int.cgenideal_self
thf(fact_1083_int_Oa__r__coset__subset__G,axiom,
! [H: set_int,X: int] :
( ( ord_less_eq_set_int @ H @ top_top_set_int )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ord_less_eq_set_int @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ H @ X ) @ top_top_set_int ) ) ) ).
% int.a_r_coset_subset_G
thf(fact_1084_int_Oa__coset__add__assoc,axiom,
! [M: set_int,G: int,H2: int] :
( ( ord_less_eq_set_int @ M @ top_top_set_int )
=> ( ( member_int @ G @ top_top_set_int )
=> ( ( member_int @ H2 @ top_top_set_int )
=> ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ G ) @ H2 )
= ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ ( plus_plus_int @ G @ H2 ) ) ) ) ) ) ).
% int.a_coset_add_assoc
thf(fact_1085_int_Oa__rcosI,axiom,
! [H2: int,H: set_int,X: int] :
( ( member_int @ H2 @ H )
=> ( ( ord_less_eq_set_int @ H @ top_top_set_int )
=> ( ( member_int @ X @ top_top_set_int )
=> ( member_int @ ( plus_plus_int @ H2 @ X ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ H @ X ) ) ) ) ) ).
% int.a_rcosI
thf(fact_1086_int_Ogenideal__self,axiom,
! [S: set_int] :
( ( ord_less_eq_set_int @ S @ top_top_set_int )
=> ( ord_less_eq_set_int @ S @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ S ) ) ) ).
% int.genideal_self
thf(fact_1087_int_Osubset__Idl__subset,axiom,
! [I3: set_int,H: set_int] :
( ( ord_less_eq_set_int @ I3 @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ H @ I3 )
=> ( ord_less_eq_set_int @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ H ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I3 ) ) ) ) ).
% int.subset_Idl_subset
thf(fact_1088_int_Ocgenideal__is__principalideal,axiom,
! [I: int] :
( ( member_int @ I @ top_top_set_int )
=> ( princi1768892856804252751t_unit @ ( cgenid7570434961818237563t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I ) @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ).
% int.cgenideal_is_principalideal
thf(fact_1089_int_OIdl__subset__ideal_H,axiom,
! [A2: int,B: int] :
( ( member_int @ A2 @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ B @ bot_bot_set_int ) ) )
= ( member_int @ A2 @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ) ) ).
% int.Idl_subset_ideal'
thf(fact_1090_int_Omaximalideal__prime,axiom,
! [I3: set_int] :
( ( maxima7040249999675607092t_unit @ I3 @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) )
=> ( primei2109666362732673920t_unit @ I3 @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ).
% int.maximalideal_prime
thf(fact_1091_int_Oa__rcosetsI,axiom,
! [H: set_int,X: int] :
( ( ord_less_eq_set_int @ H @ top_top_set_int )
=> ( ( member_int @ X @ top_top_set_int )
=> ( member_set_int @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ H @ X ) @ ( a_RCOS3445019769541752303t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ H ) ) ) ) ).
% int.a_rcosetsI
thf(fact_1092_Idl__subset__eq__dvd,axiom,
! [K: int,L: int] :
( ( ord_less_eq_set_int @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ K @ bot_bot_set_int ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ L @ bot_bot_set_int ) ) )
= ( dvd_dvd_int @ L @ K ) ) ).
% Idl_subset_eq_dvd
thf(fact_1093_int_Ole__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( X = Y ) ) ) ) ) ).
% int.le_antisym
thf(fact_1094_int_Ole__refl,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ord_less_eq_int @ X @ X ) ) ).
% int.le_refl
thf(fact_1095_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1096_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1097_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1098_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1099_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( modulo_modulo_int @ K @ L )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_1100_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( modulo_modulo_int @ K @ L )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_1101_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1102_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1103_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1104_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1105_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1106_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1107_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1108_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1109_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1110_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1111_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1112_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_1113_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_1114_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1115_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1116_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_1117_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1118_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1119_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1120_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1121_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1122_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_1123_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_1124_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_1125_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_1126_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1127_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_1128_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1129_zdvd__antisym__nonneg,axiom,
! [M2: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M2 @ N )
=> ( ( dvd_dvd_int @ N @ M2 )
=> ( M2 = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_1130_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( ( dvd_dvd_int @ Z @ N )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ Z @ N ) ) ) ).
% zdvd_imp_le
thf(fact_1131_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1132_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1133_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1134_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1135_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1136_int_Ototal__order__total,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% int.total_order_total
thf(fact_1137_int_Ole__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ord_less_eq_int @ X @ Z ) ) ) ) ) ) ).
% int.le_trans
thf(fact_1138_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_1139_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_1140_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_1141_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1142_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1143_int_Olless__eq,axiom,
( ord_less_int
= ( ^ [X3: int,Y5: int] :
( ( ord_less_eq_int @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% int.lless_eq
thf(fact_1144_mod__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
= ( ( dvd_dvd_int @ L @ K )
| ( ( L = zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ K ) )
| ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_1145_zdvd__mult__cancel,axiom,
! [K: int,M2: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M2 ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M2 @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_1146_zdvd__period,axiom,
! [A2: int,D2: int,X: int,T: int,C2: int] :
( ( dvd_dvd_int @ A2 @ D2 )
=> ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ X @ T ) )
= ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C2 @ D2 ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_1147_zdvd__reduce,axiom,
! [K: int,N: int,M2: int] :
( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M2 ) ) )
= ( dvd_dvd_int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_1148_zdvd__not__zless,axiom,
! [M2: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ord_less_int @ M2 @ N )
=> ~ ( dvd_dvd_int @ N @ M2 ) ) ) ).
% zdvd_not_zless
thf(fact_1149_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1150_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1151_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1152_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_1153_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1154_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z4: int] :
? [N4: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1155_zmod__le__nonneg__dividend,axiom,
! [M2: int,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ M2 )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ M2 @ K ) @ M2 ) ) ).
% zmod_le_nonneg_dividend
thf(fact_1156_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1157_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1158_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1159_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1160_mod__le__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_1161_mod__eq__nat2E,axiom,
! [M2: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo_nat @ M2 @ Q3 )
= ( modulo_modulo_nat @ N @ Q3 ) )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ! [S3: nat] :
( N
!= ( plus_plus_nat @ M2 @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_1162_mod__eq__nat1E,axiom,
! [M2: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo_nat @ M2 @ Q3 )
= ( modulo_modulo_nat @ N @ Q3 ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ~ ! [S3: nat] :
( M2
!= ( plus_plus_nat @ N @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_1163_zmod__trivial__iff,axiom,
! [I: int,K: int] :
( ( ( modulo_modulo_int @ I @ K )
= I )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zmod_trivial_iff
thf(fact_1164_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_1165_neg__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% neg_mod_sign
thf(fact_1166_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1167_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( modulo_modulo_int @ K @ L )
= ( plus_plus_int @ K @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_1168_int__mod__pos__eq,axiom,
! [A2: int,B: int,Q3: int,R: int] :
( ( A2
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ R @ B )
=> ( ( modulo_modulo_int @ A2 @ B )
= R ) ) ) ) ).
% int_mod_pos_eq
thf(fact_1169_int__mod__neg__eq,axiom,
! [A2: int,B: int,Q3: int,R: int] :
( ( A2
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R ) )
=> ( ( ord_less_eq_int @ R @ zero_zero_int )
=> ( ( ord_less_int @ B @ R )
=> ( ( modulo_modulo_int @ A2 @ B )
= R ) ) ) ) ).
% int_mod_neg_eq
thf(fact_1170_split__zmod,axiom,
! [Q: int > $o,N: int,K: int] :
( ( Q @ ( modulo_modulo_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( Q @ N ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I4: int,J2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
& ( ord_less_int @ J2 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J2 ) ) )
=> ( Q @ J2 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I4: int,J2: int] :
( ( ( ord_less_int @ K @ J2 )
& ( ord_less_eq_int @ J2 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J2 ) ) )
=> ( Q @ J2 ) ) ) ) ) ).
% split_zmod
thf(fact_1171_dvds__eq__Idl,axiom,
! [L: int,K: int] :
( ( ( dvd_dvd_int @ L @ K )
& ( dvd_dvd_int @ K @ L ) )
= ( ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ K @ bot_bot_set_int ) )
= ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ L @ bot_bot_set_int ) ) ) ) ).
% dvds_eq_Idl
thf(fact_1172_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D2 ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1173_nat__dvd__1__iff__1,axiom,
! [M2: nat] :
( ( dvd_dvd_nat @ M2 @ one_one_nat )
= ( M2 = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_1174_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M2 @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1175_int__dvd__int__iff,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M2 @ N ) ) ).
% int_dvd_int_iff
thf(fact_1176_verit__la__generic,axiom,
! [A2: int,X: int] :
( ( ord_less_eq_int @ A2 @ X )
| ( A2 = X )
| ( ord_less_eq_int @ X @ A2 ) ) ).
% verit_la_generic
thf(fact_1177_nat__dvd__not__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_nat @ M2 @ N )
=> ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ).
% nat_dvd_not_less
thf(fact_1178_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_1179_nat__mult__dvd__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( dvd_dvd_nat @ M2 @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_1180_dvd__mult__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M2 @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_1181_mod__greater__zero__iff__not__dvd,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ N ) )
= ( ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_1182_conj__le__cong,axiom,
! [X: int,X8: int,P: $o,P5: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_1183_imp__le__cong,axiom,
! [X: int,X8: int,P: $o,P5: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_1184_dvd__mult__cancel1,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M2 @ N ) @ M2 )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_1185_dvd__mult__cancel2,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M2 ) @ M2 )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_1186_zdvd__mono,axiom,
! [K: int,M2: int,T: int] :
( ( K != zero_zero_int )
=> ( ( dvd_dvd_int @ M2 @ T )
= ( dvd_dvd_int @ ( times_times_int @ K @ M2 ) @ ( times_times_int @ K @ T ) ) ) ) ).
% zdvd_mono
thf(fact_1187_prime__primeideal,axiom,
! [P6: int] :
( ( factor1798656936486255268me_int @ P6 )
=> ( primei2109666362732673920t_unit @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ P6 @ bot_bot_set_int ) ) @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ).
% prime_primeideal
thf(fact_1188_int_Ozeromaximalideal__fieldI,axiom,
( ( maxima7040249999675607092t_unit @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) )
=> ( field_5117527561578272769t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ).
% int.zeromaximalideal_fieldI
thf(fact_1189_dvd__antisym,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd_nat @ M2 @ N )
=> ( ( dvd_dvd_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% dvd_antisym
thf(fact_1190_int_Ocring__fieldI2,axiom,
( ( zero_zero_int != one_one_int )
=> ( ! [A4: int] :
( ( member_int @ A4 @ top_top_set_int )
=> ( ( A4 != zero_zero_int )
=> ? [X4: int] :
( ( member_int @ X4 @ top_top_set_int )
& ( ( times_times_int @ A4 @ X4 )
= one_one_int ) ) ) )
=> ( field_5117527561578272769t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ).
% int.cring_fieldI2
thf(fact_1191_int_Ozeromaximalideal__eq__field,axiom,
( ( maxima7040249999675607092t_unit @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) )
= ( field_5117527561578272769t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ).
% int.zeromaximalideal_eq_field
thf(fact_1192_Primes_Oprime__int__iff,axiom,
( factor1798656936486255268me_int
= ( ^ [N4: int] :
( ( ord_less_int @ one_one_int @ N4 )
& ! [M4: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ M4 )
& ( dvd_dvd_int @ M4 @ N4 ) )
=> ( ( M4 = one_one_int )
| ( M4 = N4 ) ) ) ) ) ) ).
% Primes.prime_int_iff
thf(fact_1193_prime__int__altdef,axiom,
( factor1798656936486255268me_int
= ( ^ [P4: int] :
( ( ord_less_int @ one_one_int @ P4 )
& ! [M4: int] :
( ( ord_less_eq_int @ zero_zero_int @ M4 )
=> ( ( dvd_dvd_int @ M4 @ P4 )
=> ( ( M4 = one_one_int )
| ( M4 = P4 ) ) ) ) ) ) ) ).
% prime_int_altdef
thf(fact_1194_prime__ge__0__int,axiom,
! [P6: int] :
( ( factor1798656936486255268me_int @ P6 )
=> ( ord_less_eq_int @ zero_zero_int @ P6 ) ) ).
% prime_ge_0_int
thf(fact_1195_prime__ge__1__int,axiom,
! [P6: int] :
( ( factor1798656936486255268me_int @ P6 )
=> ( ord_less_eq_int @ one_one_int @ P6 ) ) ).
% prime_ge_1_int
thf(fact_1196_prime__gt__0__int,axiom,
! [P6: int] :
( ( factor1798656936486255268me_int @ P6 )
=> ( ord_less_int @ zero_zero_int @ P6 ) ) ).
% prime_gt_0_int
thf(fact_1197_prime__gt__1__int,axiom,
! [P6: int] :
( ( factor1798656936486255268me_int @ P6 )
=> ( ord_less_int @ one_one_int @ P6 ) ) ).
% prime_gt_1_int
thf(fact_1198_prime__dvd__mult__eq__int,axiom,
! [P6: int,A2: int,B: int] :
( ( factor1798656936486255268me_int @ P6 )
=> ( ( dvd_dvd_int @ P6 @ ( times_times_int @ A2 @ B ) )
= ( ( dvd_dvd_int @ P6 @ A2 )
| ( dvd_dvd_int @ P6 @ B ) ) ) ) ).
% prime_dvd_mult_eq_int
thf(fact_1199_prime__int__not__dvd,axiom,
! [P6: int,N: int] :
( ( factor1798656936486255268me_int @ P6 )
=> ( ( ord_less_int @ N @ P6 )
=> ( ( ord_less_int @ one_one_int @ N )
=> ~ ( dvd_dvd_int @ N @ P6 ) ) ) ) ).
% prime_int_not_dvd
thf(fact_1200_bezout__add__strong__nat,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ? [D3: nat,X2: nat,Y2: nat] :
( ( dvd_dvd_nat @ D3 @ A2 )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( times_times_nat @ A2 @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D3 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1201_prime__nat__int__transfer,axiom,
! [N: nat] :
( ( factor1798656936486255268me_int @ ( semiri1314217659103216013at_int @ N ) )
= ( factor1801147406995305544me_nat @ N ) ) ).
% prime_nat_int_transfer
thf(fact_1202_prime__ge__1__nat,axiom,
! [P6: nat] :
( ( factor1801147406995305544me_nat @ P6 )
=> ( ord_less_eq_nat @ one_one_nat @ P6 ) ) ).
% prime_ge_1_nat
thf(fact_1203_prime__factor__nat,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ? [P7: nat] :
( ( factor1801147406995305544me_nat @ P7 )
& ( dvd_dvd_nat @ P7 @ N ) ) ) ).
% prime_factor_nat
thf(fact_1204_prime__dvd__mult__eq__nat,axiom,
! [P6: nat,A2: nat,B: nat] :
( ( factor1801147406995305544me_nat @ P6 )
=> ( ( dvd_dvd_nat @ P6 @ ( times_times_nat @ A2 @ B ) )
= ( ( dvd_dvd_nat @ P6 @ A2 )
| ( dvd_dvd_nat @ P6 @ B ) ) ) ) ).
% prime_dvd_mult_eq_nat
thf(fact_1205_bigger__prime,axiom,
! [N: nat] :
? [P7: nat] :
( ( factor1801147406995305544me_nat @ P7 )
& ( ord_less_nat @ N @ P7 ) ) ).
% bigger_prime
thf(fact_1206_prime__gt__1__nat,axiom,
! [P6: nat] :
( ( factor1801147406995305544me_nat @ P6 )
=> ( ord_less_nat @ one_one_nat @ P6 ) ) ).
% prime_gt_1_nat
thf(fact_1207_prime__gt__0__nat,axiom,
! [P6: nat] :
( ( factor1801147406995305544me_nat @ P6 )
=> ( ord_less_nat @ zero_zero_nat @ P6 ) ) ).
% prime_gt_0_nat
thf(fact_1208_prime__product,axiom,
! [P6: nat,Q3: nat] :
( ( factor1801147406995305544me_nat @ ( times_times_nat @ P6 @ Q3 ) )
=> ( ( P6 = one_one_nat )
| ( Q3 = one_one_nat ) ) ) ).
% prime_product
thf(fact_1209_not__prime__eq__prod__nat,axiom,
! [M2: nat] :
( ( ord_less_nat @ one_one_nat @ M2 )
=> ( ~ ( factor1801147406995305544me_nat @ M2 )
=> ? [N3: nat,K2: nat] :
( ( N3
= ( times_times_nat @ M2 @ K2 ) )
& ( ord_less_nat @ one_one_nat @ M2 )
& ( ord_less_nat @ M2 @ N3 )
& ( ord_less_nat @ one_one_nat @ K2 )
& ( ord_less_nat @ K2 @ N3 ) ) ) ) ).
% not_prime_eq_prod_nat
thf(fact_1210_prime__nat__iff,axiom,
( factor1801147406995305544me_nat
= ( ^ [N4: nat] :
( ( ord_less_nat @ one_one_nat @ N4 )
& ! [M4: nat] :
( ( dvd_dvd_nat @ M4 @ N4 )
=> ( ( M4 = one_one_nat )
| ( M4 = N4 ) ) ) ) ) ) ).
% prime_nat_iff
thf(fact_1211_prime__nat__not__dvd,axiom,
! [P6: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P6 )
=> ( ( ord_less_nat @ N @ P6 )
=> ( ( N != one_one_nat )
=> ~ ( dvd_dvd_nat @ N @ P6 ) ) ) ) ).
% prime_nat_not_dvd
thf(fact_1212_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A4: nat,B5: nat] :
( ( P @ A4 @ B5 )
= ( P @ B5 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B5: nat] :
( ( P @ A4 @ B5 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B5 ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1213_gcd__nat_Oextremum,axiom,
! [A2: nat] : ( dvd_dvd_nat @ A2 @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_1214_gcd__nat_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
& ( zero_zero_nat != A2 ) ) ).
% gcd_nat.extremum_strict
thf(fact_1215_gcd__nat_Oextremum__unique,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
= ( A2 = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_1216_gcd__nat_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ( dvd_dvd_nat @ A2 @ zero_zero_nat )
& ( A2 != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_1217_gcd__nat_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
=> ( A2 = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1218_dvd__pos__nat,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M2 @ N )
=> ( ord_less_nat @ zero_zero_nat @ M2 ) ) ) ).
% dvd_pos_nat
thf(fact_1219_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [M5: nat] : ( P @ M5 @ zero_zero_nat )
=> ( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 @ ( modulo_modulo_nat @ M5 @ N3 ) )
=> ( P @ M5 @ N3 ) ) )
=> ( P @ M2 @ N ) ) ) ).
% gcd_nat_induct
thf(fact_1220_bezout__add__nat,axiom,
! [A2: nat,B: nat] :
? [D3: nat,X2: nat,Y2: nat] :
( ( dvd_dvd_nat @ D3 @ A2 )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( ( times_times_nat @ A2 @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y2 ) @ D3 ) )
| ( ( times_times_nat @ B @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y2 ) @ D3 ) ) ) ) ).
% bezout_add_nat
thf(fact_1221_bezout__lemma__nat,axiom,
! [D2: nat,A2: nat,B: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ D2 @ A2 )
=> ( ( dvd_dvd_nat @ D2 @ B )
=> ( ( ( ( times_times_nat @ A2 @ X )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D2 ) )
| ( ( times_times_nat @ B @ X )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y ) @ D2 ) ) )
=> ? [X2: nat,Y2: nat] :
( ( dvd_dvd_nat @ D2 @ A2 )
& ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ A2 @ B ) )
& ( ( ( times_times_nat @ A2 @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ Y2 ) @ D2 ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ X2 )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ Y2 ) @ D2 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1222_dvd__nat__bounds,axiom,
! [P6: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ P6 )
=> ( ( dvd_dvd_nat @ N @ P6 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ord_less_eq_nat @ N @ P6 ) ) ) ) ).
% dvd_nat_bounds
thf(fact_1223_int_Ozeroprimeideal__domainI,axiom,
( ( primei2109666362732673920t_unit @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) )
=> ( domain1430183510194609567t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ).
% int.zeroprimeideal_domainI
thf(fact_1224_int_Odomain__axioms,axiom,
domain1430183510194609567t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ).
% int.domain_axioms
thf(fact_1225_int_Odomain__eq__zeroprimeideal,axiom,
( ( domain1430183510194609567t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) )
= ( primei2109666362732673920t_unit @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ).
% int.domain_eq_zeroprimeideal
thf(fact_1226_int_Ocharacteristic__is__prime,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( ring_c8451697193457390130t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( factor1801147406995305544me_nat @ ( ring_c8451697193457390130t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ).
% int.characteristic_is_prime
thf(fact_1227_Idl__eq__abs,axiom,
! [K: int,L: int] :
( ( ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ K @ bot_bot_set_int ) )
= ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ L @ bot_bot_set_int ) ) )
= ( ( abs_abs_int @ L )
= ( abs_abs_int @ K ) ) ) ).
% Idl_eq_abs
thf(fact_1228_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
= ( ( abs_abs_int @ X )
= one_one_int ) ) ).
% zdvd1_eq
thf(fact_1229_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1230_abs__zmult__eq__1,axiom,
! [M2: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M2 @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M2 )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_1231_int__in__range__abs,axiom,
! [N: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% int_in_range_abs
thf(fact_1232_zdvd__antisym__abs,axiom,
! [A2: int,B: int] :
( ( dvd_dvd_int @ A2 @ B )
=> ( ( dvd_dvd_int @ B @ A2 )
=> ( ( abs_abs_int @ A2 )
= ( abs_abs_int @ B ) ) ) ) ).
% zdvd_antisym_abs
thf(fact_1233_dvd__imp__le__int,axiom,
! [I: int,D2: int] :
( ( I != zero_zero_int )
=> ( ( dvd_dvd_int @ D2 @ I )
=> ( ord_less_eq_int @ ( abs_abs_int @ D2 ) @ ( abs_abs_int @ I ) ) ) ) ).
% dvd_imp_le_int
thf(fact_1234_abs__mod__less,axiom,
! [L: int,K: int] :
( ( L != zero_zero_int )
=> ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% abs_mod_less
thf(fact_1235_prime__factor__int,axiom,
! [K: int] :
( ( ( abs_abs_int @ K )
!= one_one_int )
=> ~ ! [P7: int] :
( ( factor1798656936486255268me_int @ P7 )
=> ~ ( dvd_dvd_int @ P7 @ K ) ) ) ).
% prime_factor_int
thf(fact_1236_zdvd__mult__cancel1,axiom,
! [M2: int,N: int] :
( ( M2 != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ M2 @ N ) @ M2 )
= ( ( abs_abs_int @ N )
= one_one_int ) ) ) ).
% zdvd_mult_cancel1
thf(fact_1237_int_Ochar__bound_I1_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( semiri1314217659103216013at_int @ X ) )
= zero_zero_int )
=> ( ord_less_eq_nat @ ( ring_c8451697193457390130t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) @ X ) ) ) ).
% int.char_bound(1)
thf(fact_1238_int_Ofinite__carr__imp__char__ge__0,axiom,
( ( finite_finite_int @ top_top_set_int )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c8451697193457390130t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ).
% int.finite_carr_imp_char_ge_0
thf(fact_1239_infinite__UNIV__int,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_int
thf(fact_1240_ZFact__prime__is__domain,axiom,
! [P6: int] :
( ( factor1798656936486255268me_int @ P6 )
=> ( domain6183376680155302761t_unit @ ( zFact @ P6 ) ) ) ).
% ZFact_prime_is_domain
thf(fact_1241_int_Oint__embed__add,axiom,
! [X: int,Y: int] :
( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X ) @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y ) ) ) ).
% int.int_embed_add
thf(fact_1242_int_Oint__embed__one,axiom,
( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ one_one_int )
= one_one_int ) ).
% int.int_embed_one
thf(fact_1243_int_Oint__embed__mult,axiom,
! [X: int,Y: int] :
( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( times_times_int @ X @ Y ) )
= ( times_times_int @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X ) @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y ) ) ) ).
% int.int_embed_mult
thf(fact_1244_int_Oint__embed__zero,axiom,
( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ zero_zero_int )
= zero_zero_int ) ).
% int.int_embed_zero
thf(fact_1245_int_Oint__embed__closed,axiom,
! [K: int] : ( member_int @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K ) @ top_top_set_int ) ).
% int.int_embed_closed
thf(fact_1246_int_Oint__embed__mult__aux,axiom,
! [X: int,Y: nat] :
( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( times_times_int @ X @ ( semiri1314217659103216013at_int @ Y ) ) )
= ( times_times_int @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X ) @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).
% int.int_embed_mult_aux
thf(fact_1247_int_Oembed__char__eq__0,axiom,
( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( semiri1314217659103216013at_int @ ( ring_c8451697193457390130t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) )
= zero_zero_int ) ).
% int.embed_char_eq_0
thf(fact_1248_int_Oembed__char__eq__0__iff,axiom,
! [N: int] :
( ( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N )
= zero_zero_int )
= ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ ( ring_c8451697193457390130t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) @ N ) ) ).
% int.embed_char_eq_0_iff
thf(fact_1249_int_Ochar__bound_I2_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( semiri1314217659103216013at_int @ X ) )
= zero_zero_int )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c8451697193457390130t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ) ).
% int.char_bound(2)
thf(fact_1250_int_Oorder__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_3730749346064586230t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
= ( finite_finite_int @ top_top_set_int ) ) ).
% int.order_gt_0_iff_finite
thf(fact_1251_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_1252_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1253_bounded__nat__set__is__finite,axiom,
! [N2: set_nat,N: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ord_less_nat @ X2 @ N ) )
=> ( finite_finite_nat @ N2 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1254_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1255_fin__zfact,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% fin_zfact
thf(fact_1256_finite__int__iff__bounded,axiom,
( finite_finite_int
= ( ^ [S4: set_int] :
? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S4 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).
% finite_int_iff_bounded
thf(fact_1257_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_1258_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1259_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M5: nat] :
( ( ord_less_nat @ K @ M5 )
=> ? [N6: nat] :
( ( ord_less_nat @ M5 @ N6 )
& ( member_nat @ N6 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_1260_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M4: nat] :
? [N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ( member_nat @ N4 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1261_finite__nat__bounded,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ? [K2: nat] : ( ord_less_eq_set_nat @ S @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% finite_nat_bounded
thf(fact_1262_finite__nat__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [S4: set_nat] :
? [K3: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_1263_infinite__int__iff__unbounded,axiom,
! [S: set_int] :
( ( ~ ( finite_finite_int @ S ) )
= ( ! [M4: int] :
? [N4: int] :
( ( ord_less_int @ M4 @ ( abs_abs_int @ N4 ) )
& ( member_int @ N4 @ S ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_1264_int_Ochar__ring__is__subdomain,axiom,
subdom8488461989912461802t_unit @ ( image_int_int @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) @ top_top_set_int ) @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ).
% int.char_ring_is_subdomain
thf(fact_1265_int_Ochar__ring__is__subfield,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( ring_c8451697193457390130t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
=> ( subfie2008777110680908022t_unit @ ( image_int_int @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) @ top_top_set_int ) @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ).
% int.char_ring_is_subfield
thf(fact_1266_int_Oquot__domain__imp__primeideal,axiom,
! [P: set_int] :
( ( ideal_6787631597145370931t_unit @ P @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) )
=> ( ( domain6183376680155302761t_unit @ ( factRi5755170488246124606t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ P ) )
=> ( primei2109666362732673920t_unit @ P @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ).
% int.quot_domain_imp_primeideal
thf(fact_1267_int_Oquot__domain__iff__primeideal,axiom,
! [P: set_int] :
( ( ideal_6787631597145370931t_unit @ P @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) )
=> ( ( domain6183376680155302761t_unit @ ( factRi5755170488246124606t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ P ) )
= ( primei2109666362732673920t_unit @ P @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ).
% int.quot_domain_iff_primeideal
% 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_set_int @ x @ ( image_nat_set_int @ ( ring_zfact_iso @ n ) @ ( set_ord_lessThan_nat @ n ) ) ).
%------------------------------------------------------------------------------