TPTP Problem File: SLH0196^1.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_00079_002494__18239784_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1433 ( 530 unt; 162 typ; 0 def)
% Number of atoms : 3551 (1200 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 11367 ( 311 ~; 86 |; 207 &;9387 @)
% ( 0 <=>;1376 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 18 ( 17 usr)
% Number of type conns : 397 ( 397 >; 0 *; 0 +; 0 <<)
% Number of symbols : 146 ( 145 usr; 19 con; 0-4 aty)
% Number of variables : 2860 ( 87 ^;2645 !; 128 ?;2860 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:40:26.279
%------------------------------------------------------------------------------
% Could-be-implicit typings (17)
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% Explicit typings (145)
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ord_le5995675665013768039et_int: set_nat_set_int > set_nat_set_int > $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_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
ord_le4317611570275147438et_int: set_set_set_int > set_set_set_int > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
top_to6689522016564766135et_int: set_nat_set_int ).
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_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
top_to5524576366173240574et_int: set_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_OFactRing_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
factRi3149420076008518152t_unit: partia4934656038542163276t_unit > set_set_int > partia3601206958761670294t_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_QuotRing_Ois__ring__iso_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
is_rin4964436081320486288t_unit: partia4934656038542163276t_unit > partia3601206958761670294t_unit > $o ).
thf(sy_c_QuotRing_Ois__ring__iso_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
is_rin1729965262454186108t_unit: partia3601206958761670294t_unit > partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Oa__minus_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
a_minu5974516859897376926t_unit: partia4934656038542163276t_unit > set_int > set_int > set_int ).
thf(sy_c_Ring_Oabelian__monoid_001t__Int__Oint_001t__Product____Type__Ounit,type,
abelia6702305049305627311t_unit: partia2818514838349642498t_unit > $o ).
thf(sy_c_Ring_Oabelian__monoid_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
abelia3815030880812984441t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Oabelian__monoid_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
abelia4259588778567990595t_unit: partia3601206958761670294t_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_Odomain_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
domain2593739924244766771t_unit: partia3601206958761670294t_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_Ofield_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
field_5943785737635511755t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
field_4969826516285655701t_unit: partia3601206958761670294t_unit > $o ).
thf(sy_c_Ring_Oring_Oadd_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
add_se5859248395121729892t_unit: partia4934656038542163276t_unit > set_int > set_int > set_int ).
thf(sy_c_Ring_Oring_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_Oring_Oring__ext_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_e3077432647605977236t_unit: set_int > ( set_int > set_int > set_int ) > product_unit > ring_e183874169221349138t_unit ).
thf(sy_c_Ring_Oring_Ozero_001t__Int__Oint_001t__Product____Type__Ounit,type,
zero_i2266321264637750939t_unit: partia2818514838349642498t_unit > int ).
thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
zero_s6269048424454532197t_unit: partia4934656038542163276t_unit > set_int ).
thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
zero_s2946895028292161839t_unit: partia3601206958761670294t_unit > set_set_int ).
thf(sy_c_Ring_Osemiring_001t__Int__Oint_001t__Product____Type__Ounit,type,
semiri1037594922888297541t_unit: partia2818514838349642498t_unit > $o ).
thf(sy_c_Ring_Osemiring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
semiri8708897239777792527t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Osemiring_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
semiri9009874074179768025t_unit: partia3601206958761670294t_unit > $o ).
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_Set_OCollect_001_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
collect_nat_set_int: ( ( nat > set_int ) > $o ) > set_nat_set_int ).
thf(sy_c_Set_OCollect_001t__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_OCollect_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
collect_set_set_int: ( set_set_int > $o ) > set_set_set_int ).
thf(sy_c_Set_OPow_001t__Int__Oint,type,
pow_int: set_int > set_set_int ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Int__Oint_J,type,
pow_set_int: set_set_int > set_set_set_int ).
thf(sy_c_Set_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__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
image_1010086626112315521et_int: ( set_int > set_set_int ) > set_set_int > set_set_set_int ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
insert_nat_set_int: ( nat > set_int ) > set_nat_set_int > set_nat_set_int ).
thf(sy_c_Set_Oinsert_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_Oinsert_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
insert_set_set_int: set_set_int > set_set_set_int > set_set_set_int ).
thf(sy_c_UnivPoly_Obound_001t__Set__Oset_It__Int__Oint_J,type,
bound_set_int: set_int > nat > ( nat > set_int ) > $o ).
thf(sy_c_UnivPoly_Oup_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
up_set1168727741560211120t_unit: partia4934656038542163276t_unit > set_nat_set_int ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Set__Oset_It__Int__Oint_J_J,type,
member_nat_set_int: ( nat > set_int ) > set_nat_set_int > $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_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
member_set_set_int: set_set_int > set_set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
member7356822600254261989et_int: set_set_set_int > set_set_set_set_int > $o ).
thf(sy_v_I____,type,
i: set_int ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1270)
thf(fact_0_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_1_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_2_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_3_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_4_int_Ozeroideal,axiom,
ideal_6787631597145370931t_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.zeroideal
thf(fact_5_int_Ooneideal,axiom,
ideal_6787631597145370931t_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.oneideal
thf(fact_6_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_7_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_8_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_9_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_10_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_11_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_12_int_Oadd_Ol__cancel__one,axiom,
! [X: int,A: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( ( plus_plus_int @ X @ A )
= X )
= ( A = zero_zero_int ) ) ) ) ).
% int.add.l_cancel_one
thf(fact_13_UNIV_I3_J,axiom,
! [P: int > $o] :
( ( ! [X2: int] :
( ( member_int @ X2 @ top_top_set_int )
=> ( P @ X2 ) ) )
= ( ! [X3: int] : ( P @ X3 ) ) ) ).
% UNIV(3)
thf(fact_14_UNIV_I4_J,axiom,
! [P: int > $o] :
( ( ? [X2: int] :
( ( member_int @ X2 @ top_top_set_int )
& ( P @ X2 ) ) )
= ( ? [X3: int] : ( P @ X3 ) ) ) ).
% UNIV(4)
thf(fact_15_int_Oadd_Oone__closed,axiom,
member_int @ zero_zero_int @ top_top_set_int ).
% int.add.one_closed
thf(fact_16_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_17_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_18_int_Oone__closed,axiom,
member_int @ one_one_int @ top_top_set_int ).
% int.one_closed
thf(fact_19_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_20_int_Oadd_Or__cancel__one_H,axiom,
! [X: int,A: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( X
= ( plus_plus_int @ A @ X ) )
= ( A = zero_zero_int ) ) ) ) ).
% int.add.r_cancel_one'
thf(fact_21_int_Oadd_Ol__cancel__one_H,axiom,
! [X: int,A: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( X
= ( plus_plus_int @ X @ A ) )
= ( A = zero_zero_int ) ) ) ) ).
% int.add.l_cancel_one'
thf(fact_22_int_Oadd_Or__cancel__one,axiom,
! [X: int,A: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( ( plus_plus_int @ A @ X )
= X )
= ( A = zero_zero_int ) ) ) ) ).
% int.add.r_cancel_one
thf(fact_23_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_24_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_25_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_26_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_27_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_28_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_29_int_Ozero__not__one,axiom,
zero_zero_int != one_one_int ).
% int.zero_not_one
thf(fact_30_int_Oadd_Ocarrier__not__empty,axiom,
top_top_set_int != bot_bot_set_int ).
% int.add.carrier_not_empty
thf(fact_31_int_Oadd_Oone__unique,axiom,
! [U: int] :
( ( member_int @ U @ top_top_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ top_top_set_int )
=> ( ( plus_plus_int @ U @ X4 )
= X4 ) )
=> ( U = zero_zero_int ) ) ) ).
% int.add.one_unique
thf(fact_32_int_Oadd_Oinv__unique,axiom,
! [Y: int,X: int,Y2: int] :
( ( ( plus_plus_int @ Y @ X )
= zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y2 )
= zero_zero_int )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Y2 @ top_top_set_int )
=> ( Y = Y2 ) ) ) ) ) ) ).
% int.add.inv_unique
thf(fact_33_int_Oadd_Or__inv__ex,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ? [X4: int] :
( ( member_int @ X4 @ top_top_set_int )
& ( ( plus_plus_int @ X @ X4 )
= zero_zero_int ) ) ) ).
% int.add.r_inv_ex
thf(fact_34_int_Oadd_Ol__inv__ex,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ? [X4: int] :
( ( member_int @ X4 @ top_top_set_int )
& ( ( plus_plus_int @ X4 @ X )
= zero_zero_int ) ) ) ).
% int.add.l_inv_ex
thf(fact_35_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_36_int_Ointegral__iff,axiom,
! [A: int,B: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ) ) ).
% int.integral_iff
thf(fact_37_int_Om__rcancel,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C @ top_top_set_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% int.m_rcancel
thf(fact_38_int_Om__lcancel,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C @ top_top_set_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% int.m_lcancel
thf(fact_39_int_Ointegral,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ) ) ).
% int.integral
thf(fact_40_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_41_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_42_int_Oone__unique,axiom,
! [U: int] :
( ( member_int @ U @ top_top_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ top_top_set_int )
=> ( ( times_times_int @ U @ X4 )
= X4 ) )
=> ( U = one_one_int ) ) ) ).
% int.one_unique
thf(fact_43_int_Oinv__unique,axiom,
! [Y: int,X: int,Y2: int] :
( ( ( times_times_int @ Y @ X )
= one_one_int )
=> ( ( ( times_times_int @ X @ Y2 )
= one_one_int )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Y2 @ top_top_set_int )
=> ( Y = Y2 ) ) ) ) ) ) ).
% int.inv_unique
thf(fact_44_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_45_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A: set_int,P: set_int > $o] :
( ( member_set_int @ A @ ( collect_set_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: set_set_int,P: set_set_int > $o] :
( ( member_set_set_int @ A @ ( collect_set_set_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
! [A: nat > set_int,P: ( nat > set_int ) > $o] :
( ( member_nat_set_int @ A @ ( collect_nat_set_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A2: set_set_int] :
( ( collect_set_int
@ ^ [X2: set_int] : ( member_set_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A2: set_set_set_int] :
( ( collect_set_set_int
@ ^ [X2: set_set_int] : ( member_set_set_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A2: set_nat_set_int] :
( ( collect_nat_set_int
@ ^ [X2: nat > set_int] : ( member_nat_set_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_53_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_54_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_55_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_56_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_57_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_58_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_59_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_60_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_61_singletonI,axiom,
! [A: set_set_int] : ( member_set_set_int @ A @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) ).
% singletonI
thf(fact_62_singletonI,axiom,
! [A: nat > set_int] : ( member_nat_set_int @ A @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ) ).
% singletonI
thf(fact_63_singletonI,axiom,
! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% singletonI
thf(fact_64_singletonI,axiom,
! [A: set_int] : ( member_set_int @ A @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ).
% singletonI
thf(fact_65_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_66_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_67_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_68_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_69_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_70_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_71_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_72_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_73_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_74_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_75_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_76_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_77_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_78_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_79_UNIV__I,axiom,
! [X: set_int] : ( member_set_int @ X @ top_top_set_set_int ) ).
% UNIV_I
thf(fact_80_UNIV__I,axiom,
! [X: set_set_int] : ( member_set_set_int @ X @ top_to5524576366173240574et_int ) ).
% UNIV_I
thf(fact_81_UNIV__I,axiom,
! [X: nat > set_int] : ( member_nat_set_int @ X @ top_to6689522016564766135et_int ) ).
% UNIV_I
thf(fact_82_UNIV__I,axiom,
! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% UNIV_I
thf(fact_83_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_84_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_85_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_86_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_87_empty__iff,axiom,
! [C: set_set_int] :
~ ( member_set_set_int @ C @ bot_bo2384636101374064866et_int ) ).
% empty_iff
thf(fact_88_empty__iff,axiom,
! [C: nat > set_int] :
~ ( member_nat_set_int @ C @ bot_bo8417611410066262939et_int ) ).
% empty_iff
thf(fact_89_empty__iff,axiom,
! [C: int] :
~ ( member_int @ C @ bot_bot_set_int ) ).
% empty_iff
thf(fact_90_empty__iff,axiom,
! [C: set_int] :
~ ( member_set_int @ C @ bot_bot_set_set_int ) ).
% empty_iff
thf(fact_91_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_92_all__not__in__conv,axiom,
! [A2: set_set_set_int] :
( ( ! [X2: set_set_int] :
~ ( member_set_set_int @ X2 @ A2 ) )
= ( A2 = bot_bo2384636101374064866et_int ) ) ).
% all_not_in_conv
thf(fact_93_all__not__in__conv,axiom,
! [A2: set_nat_set_int] :
( ( ! [X2: nat > set_int] :
~ ( member_nat_set_int @ X2 @ A2 ) )
= ( A2 = bot_bo8417611410066262939et_int ) ) ).
% all_not_in_conv
thf(fact_94_all__not__in__conv,axiom,
! [A2: set_int] :
( ( ! [X2: int] :
~ ( member_int @ X2 @ A2 ) )
= ( A2 = bot_bot_set_int ) ) ).
% all_not_in_conv
thf(fact_95_all__not__in__conv,axiom,
! [A2: set_set_int] :
( ( ! [X2: set_int] :
~ ( member_set_int @ X2 @ A2 ) )
= ( A2 = bot_bot_set_set_int ) ) ).
% all_not_in_conv
thf(fact_96_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_97_Collect__empty__eq,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( ! [X2: int] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_98_Collect__empty__eq,axiom,
! [P: set_int > $o] :
( ( ( collect_set_int @ P )
= bot_bot_set_set_int )
= ( ! [X2: set_int] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_99_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_100_empty__Collect__eq,axiom,
! [P: int > $o] :
( ( bot_bot_set_int
= ( collect_int @ P ) )
= ( ! [X2: int] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_101_empty__Collect__eq,axiom,
! [P: set_int > $o] :
( ( bot_bot_set_set_int
= ( collect_set_int @ P ) )
= ( ! [X2: set_int] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_102_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_103_insertCI,axiom,
! [A: int,B2: set_int,B: int] :
( ( ~ ( member_int @ A @ B2 )
=> ( A = B ) )
=> ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% insertCI
thf(fact_104_insertCI,axiom,
! [A: set_int,B2: set_set_int,B: set_int] :
( ( ~ ( member_set_int @ A @ B2 )
=> ( A = B ) )
=> ( member_set_int @ A @ ( insert_set_int @ B @ B2 ) ) ) ).
% insertCI
thf(fact_105_insertCI,axiom,
! [A: set_set_int,B2: set_set_set_int,B: set_set_int] :
( ( ~ ( member_set_set_int @ A @ B2 )
=> ( A = B ) )
=> ( member_set_set_int @ A @ ( insert_set_set_int @ B @ B2 ) ) ) ).
% insertCI
thf(fact_106_insertCI,axiom,
! [A: nat > set_int,B2: set_nat_set_int,B: nat > set_int] :
( ( ~ ( member_nat_set_int @ A @ B2 )
=> ( A = B ) )
=> ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ B2 ) ) ) ).
% insertCI
thf(fact_107_insert__iff,axiom,
! [A: int,B: int,A2: set_int] :
( ( member_int @ A @ ( insert_int @ B @ A2 ) )
= ( ( A = B )
| ( member_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_108_insert__iff,axiom,
! [A: set_int,B: set_int,A2: set_set_int] :
( ( member_set_int @ A @ ( insert_set_int @ B @ A2 ) )
= ( ( A = B )
| ( member_set_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_109_insert__iff,axiom,
! [A: set_set_int,B: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ A @ ( insert_set_set_int @ B @ A2 ) )
= ( ( A = B )
| ( member_set_set_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_110_insert__iff,axiom,
! [A: nat > set_int,B: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_set_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_111_insert__absorb2,axiom,
! [X: int,A2: set_int] :
( ( insert_int @ X @ ( insert_int @ X @ A2 ) )
= ( insert_int @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_112_insert__absorb2,axiom,
! [X: set_int,A2: set_set_int] :
( ( insert_set_int @ X @ ( insert_set_int @ X @ A2 ) )
= ( insert_set_int @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_113_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_114_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_115_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_116_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_117_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_118_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_119_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_120_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_121_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_122_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_123_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_124_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_125_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_126_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_127_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_128_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_129_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_130_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_131_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_132_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_133_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_134_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_135_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_136_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_137_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_138_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_139_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_140_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_141_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_142_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_143_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_144_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_145_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_146_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_147_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_148_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_149_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_150_n__ge__1,axiom,
ord_less_nat @ one_one_nat @ n ).
% n_ge_1
thf(fact_151_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_152_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_153_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_154_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_155_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_156_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_157_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_158_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_159_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_160_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_161_UNIV__eq__I,axiom,
! [A2: set_set_int] :
( ! [X4: set_int] : ( member_set_int @ X4 @ A2 )
=> ( top_top_set_set_int = A2 ) ) ).
% UNIV_eq_I
thf(fact_162_UNIV__eq__I,axiom,
! [A2: set_set_set_int] :
( ! [X4: set_set_int] : ( member_set_set_int @ X4 @ A2 )
=> ( top_to5524576366173240574et_int = A2 ) ) ).
% UNIV_eq_I
thf(fact_163_UNIV__eq__I,axiom,
! [A2: set_nat_set_int] :
( ! [X4: nat > set_int] : ( member_nat_set_int @ X4 @ A2 )
=> ( top_to6689522016564766135et_int = A2 ) ) ).
% UNIV_eq_I
thf(fact_164_UNIV__eq__I,axiom,
! [A2: set_int] :
( ! [X4: int] : ( member_int @ X4 @ A2 )
=> ( top_top_set_int = A2 ) ) ).
% UNIV_eq_I
thf(fact_165_UNIV__witness,axiom,
? [X4: set_int] : ( member_set_int @ X4 @ top_top_set_set_int ) ).
% UNIV_witness
thf(fact_166_UNIV__witness,axiom,
? [X4: set_set_int] : ( member_set_set_int @ X4 @ top_to5524576366173240574et_int ) ).
% UNIV_witness
thf(fact_167_UNIV__witness,axiom,
? [X4: nat > set_int] : ( member_nat_set_int @ X4 @ top_to6689522016564766135et_int ) ).
% UNIV_witness
thf(fact_168_UNIV__witness,axiom,
? [X4: int] : ( member_int @ X4 @ top_top_set_int ) ).
% UNIV_witness
thf(fact_169_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_170_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_171_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_172_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_173_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_174_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_175_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_176_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_177_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_178_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_179_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_180_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_181_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_182_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_183_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_184_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_185_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_186_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_187_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_188_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_189_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_190_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_191_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_192_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_193_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_194_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_195_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_196_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_197_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_198_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_199_emptyE,axiom,
! [A: set_set_int] :
~ ( member_set_set_int @ A @ bot_bo2384636101374064866et_int ) ).
% emptyE
thf(fact_200_emptyE,axiom,
! [A: nat > set_int] :
~ ( member_nat_set_int @ A @ bot_bo8417611410066262939et_int ) ).
% emptyE
thf(fact_201_emptyE,axiom,
! [A: int] :
~ ( member_int @ A @ bot_bot_set_int ) ).
% emptyE
thf(fact_202_emptyE,axiom,
! [A: set_int] :
~ ( member_set_int @ A @ bot_bot_set_set_int ) ).
% emptyE
thf(fact_203_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_204_equals0D,axiom,
! [A2: set_set_set_int,A: set_set_int] :
( ( A2 = bot_bo2384636101374064866et_int )
=> ~ ( member_set_set_int @ A @ A2 ) ) ).
% equals0D
thf(fact_205_equals0D,axiom,
! [A2: set_nat_set_int,A: nat > set_int] :
( ( A2 = bot_bo8417611410066262939et_int )
=> ~ ( member_nat_set_int @ A @ A2 ) ) ).
% equals0D
thf(fact_206_equals0D,axiom,
! [A2: set_int,A: int] :
( ( A2 = bot_bot_set_int )
=> ~ ( member_int @ A @ A2 ) ) ).
% equals0D
thf(fact_207_equals0D,axiom,
! [A2: set_set_int,A: set_int] :
( ( A2 = bot_bot_set_set_int )
=> ~ ( member_set_int @ A @ A2 ) ) ).
% equals0D
thf(fact_208_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_209_equals0I,axiom,
! [A2: set_set_set_int] :
( ! [Y3: set_set_int] :
~ ( member_set_set_int @ Y3 @ A2 )
=> ( A2 = bot_bo2384636101374064866et_int ) ) ).
% equals0I
thf(fact_210_equals0I,axiom,
! [A2: set_nat_set_int] :
( ! [Y3: nat > set_int] :
~ ( member_nat_set_int @ Y3 @ A2 )
=> ( A2 = bot_bo8417611410066262939et_int ) ) ).
% equals0I
thf(fact_211_equals0I,axiom,
! [A2: set_int] :
( ! [Y3: int] :
~ ( member_int @ Y3 @ A2 )
=> ( A2 = bot_bot_set_int ) ) ).
% equals0I
thf(fact_212_equals0I,axiom,
! [A2: set_set_int] :
( ! [Y3: set_int] :
~ ( member_set_int @ Y3 @ A2 )
=> ( A2 = bot_bot_set_set_int ) ) ).
% equals0I
thf(fact_213_equals0I,axiom,
! [A2: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_214_ex__in__conv,axiom,
! [A2: set_set_set_int] :
( ( ? [X2: set_set_int] : ( member_set_set_int @ X2 @ A2 ) )
= ( A2 != bot_bo2384636101374064866et_int ) ) ).
% ex_in_conv
thf(fact_215_ex__in__conv,axiom,
! [A2: set_nat_set_int] :
( ( ? [X2: nat > set_int] : ( member_nat_set_int @ X2 @ A2 ) )
= ( A2 != bot_bo8417611410066262939et_int ) ) ).
% ex_in_conv
thf(fact_216_ex__in__conv,axiom,
! [A2: set_int] :
( ( ? [X2: int] : ( member_int @ X2 @ A2 ) )
= ( A2 != bot_bot_set_int ) ) ).
% ex_in_conv
thf(fact_217_ex__in__conv,axiom,
! [A2: set_set_int] :
( ( ? [X2: set_int] : ( member_set_int @ X2 @ A2 ) )
= ( A2 != bot_bot_set_set_int ) ) ).
% ex_in_conv
thf(fact_218_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_219_insertE,axiom,
! [A: int,B: int,A2: set_int] :
( ( member_int @ A @ ( insert_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_220_insertE,axiom,
! [A: set_int,B: set_int,A2: set_set_int] :
( ( member_set_int @ A @ ( insert_set_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_221_insertE,axiom,
! [A: set_set_int,B: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ A @ ( insert_set_set_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_set_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_222_insertE,axiom,
! [A: nat > set_int,B: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat_set_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_223_insertI1,axiom,
! [A: int,B2: set_int] : ( member_int @ A @ ( insert_int @ A @ B2 ) ) ).
% insertI1
thf(fact_224_insertI1,axiom,
! [A: set_int,B2: set_set_int] : ( member_set_int @ A @ ( insert_set_int @ A @ B2 ) ) ).
% insertI1
thf(fact_225_insertI1,axiom,
! [A: set_set_int,B2: set_set_set_int] : ( member_set_set_int @ A @ ( insert_set_set_int @ A @ B2 ) ) ).
% insertI1
thf(fact_226_insertI1,axiom,
! [A: nat > set_int,B2: set_nat_set_int] : ( member_nat_set_int @ A @ ( insert_nat_set_int @ A @ B2 ) ) ).
% insertI1
thf(fact_227_insertI2,axiom,
! [A: int,B2: set_int,B: int] :
( ( member_int @ A @ B2 )
=> ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% insertI2
thf(fact_228_insertI2,axiom,
! [A: set_int,B2: set_set_int,B: set_int] :
( ( member_set_int @ A @ B2 )
=> ( member_set_int @ A @ ( insert_set_int @ B @ B2 ) ) ) ).
% insertI2
thf(fact_229_insertI2,axiom,
! [A: set_set_int,B2: set_set_set_int,B: set_set_int] :
( ( member_set_set_int @ A @ B2 )
=> ( member_set_set_int @ A @ ( insert_set_set_int @ B @ B2 ) ) ) ).
% insertI2
thf(fact_230_insertI2,axiom,
! [A: nat > set_int,B2: set_nat_set_int,B: nat > set_int] :
( ( member_nat_set_int @ A @ B2 )
=> ( member_nat_set_int @ A @ ( insert_nat_set_int @ B @ B2 ) ) ) ).
% insertI2
thf(fact_231_Set_Oset__insert,axiom,
! [X: int,A2: set_int] :
( ( member_int @ X @ A2 )
=> ~ ! [B4: set_int] :
( ( A2
= ( insert_int @ X @ B4 ) )
=> ( member_int @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_232_Set_Oset__insert,axiom,
! [X: set_int,A2: set_set_int] :
( ( member_set_int @ X @ A2 )
=> ~ ! [B4: set_set_int] :
( ( A2
= ( insert_set_int @ X @ B4 ) )
=> ( member_set_int @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_233_Set_Oset__insert,axiom,
! [X: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ X @ A2 )
=> ~ ! [B4: set_set_set_int] :
( ( A2
= ( insert_set_set_int @ X @ B4 ) )
=> ( member_set_set_int @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_234_Set_Oset__insert,axiom,
! [X: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ X @ A2 )
=> ~ ! [B4: set_nat_set_int] :
( ( A2
= ( insert_nat_set_int @ X @ B4 ) )
=> ( member_nat_set_int @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_235_insert__ident,axiom,
! [X: int,A2: set_int,B2: set_int] :
( ~ ( member_int @ X @ A2 )
=> ( ~ ( member_int @ X @ B2 )
=> ( ( ( insert_int @ X @ A2 )
= ( insert_int @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_236_insert__ident,axiom,
! [X: set_int,A2: set_set_int,B2: set_set_int] :
( ~ ( member_set_int @ X @ A2 )
=> ( ~ ( member_set_int @ X @ B2 )
=> ( ( ( insert_set_int @ X @ A2 )
= ( insert_set_int @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_237_insert__ident,axiom,
! [X: set_set_int,A2: set_set_set_int,B2: set_set_set_int] :
( ~ ( member_set_set_int @ X @ A2 )
=> ( ~ ( member_set_set_int @ X @ B2 )
=> ( ( ( insert_set_set_int @ X @ A2 )
= ( insert_set_set_int @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_238_insert__ident,axiom,
! [X: nat > set_int,A2: set_nat_set_int,B2: set_nat_set_int] :
( ~ ( member_nat_set_int @ X @ A2 )
=> ( ~ ( member_nat_set_int @ X @ B2 )
=> ( ( ( insert_nat_set_int @ X @ A2 )
= ( insert_nat_set_int @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_239_insert__absorb,axiom,
! [A: int,A2: set_int] :
( ( member_int @ A @ A2 )
=> ( ( insert_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_240_insert__absorb,axiom,
! [A: set_int,A2: set_set_int] :
( ( member_set_int @ A @ A2 )
=> ( ( insert_set_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_241_insert__absorb,axiom,
! [A: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ A @ A2 )
=> ( ( insert_set_set_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_242_insert__absorb,axiom,
! [A: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ A2 )
=> ( ( insert_nat_set_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_243_insert__eq__iff,axiom,
! [A: int,A2: set_int,B: int,B2: set_int] :
( ~ ( member_int @ A @ A2 )
=> ( ~ ( member_int @ B @ B2 )
=> ( ( ( insert_int @ A @ A2 )
= ( insert_int @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C2: set_int] :
( ( A2
= ( insert_int @ B @ C2 ) )
& ~ ( member_int @ B @ C2 )
& ( B2
= ( insert_int @ A @ C2 ) )
& ~ ( member_int @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_244_insert__eq__iff,axiom,
! [A: set_int,A2: set_set_int,B: set_int,B2: set_set_int] :
( ~ ( member_set_int @ A @ A2 )
=> ( ~ ( member_set_int @ B @ B2 )
=> ( ( ( insert_set_int @ A @ A2 )
= ( insert_set_int @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C2: set_set_int] :
( ( A2
= ( insert_set_int @ B @ C2 ) )
& ~ ( member_set_int @ B @ C2 )
& ( B2
= ( insert_set_int @ A @ C2 ) )
& ~ ( member_set_int @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_245_insert__eq__iff,axiom,
! [A: set_set_int,A2: set_set_set_int,B: set_set_int,B2: set_set_set_int] :
( ~ ( member_set_set_int @ A @ A2 )
=> ( ~ ( member_set_set_int @ B @ B2 )
=> ( ( ( insert_set_set_int @ A @ A2 )
= ( insert_set_set_int @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C2: set_set_set_int] :
( ( A2
= ( insert_set_set_int @ B @ C2 ) )
& ~ ( member_set_set_int @ B @ C2 )
& ( B2
= ( insert_set_set_int @ A @ C2 ) )
& ~ ( member_set_set_int @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_246_insert__eq__iff,axiom,
! [A: nat > set_int,A2: set_nat_set_int,B: nat > set_int,B2: set_nat_set_int] :
( ~ ( member_nat_set_int @ A @ A2 )
=> ( ~ ( member_nat_set_int @ B @ B2 )
=> ( ( ( insert_nat_set_int @ A @ A2 )
= ( insert_nat_set_int @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C2: set_nat_set_int] :
( ( A2
= ( insert_nat_set_int @ B @ C2 ) )
& ~ ( member_nat_set_int @ B @ C2 )
& ( B2
= ( insert_nat_set_int @ A @ C2 ) )
& ~ ( member_nat_set_int @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_247_insert__commute,axiom,
! [X: int,Y: int,A2: set_int] :
( ( insert_int @ X @ ( insert_int @ Y @ A2 ) )
= ( insert_int @ Y @ ( insert_int @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_248_insert__commute,axiom,
! [X: set_int,Y: set_int,A2: set_set_int] :
( ( insert_set_int @ X @ ( insert_set_int @ Y @ A2 ) )
= ( insert_set_int @ Y @ ( insert_set_int @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_249_mk__disjoint__insert,axiom,
! [A: int,A2: set_int] :
( ( member_int @ A @ A2 )
=> ? [B4: set_int] :
( ( A2
= ( insert_int @ A @ B4 ) )
& ~ ( member_int @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_250_mk__disjoint__insert,axiom,
! [A: set_int,A2: set_set_int] :
( ( member_set_int @ A @ A2 )
=> ? [B4: set_set_int] :
( ( A2
= ( insert_set_int @ A @ B4 ) )
& ~ ( member_set_int @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_251_mk__disjoint__insert,axiom,
! [A: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ A @ A2 )
=> ? [B4: set_set_set_int] :
( ( A2
= ( insert_set_set_int @ A @ B4 ) )
& ~ ( member_set_set_int @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_252_mk__disjoint__insert,axiom,
! [A: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ A @ A2 )
=> ? [B4: set_nat_set_int] :
( ( A2
= ( insert_nat_set_int @ A @ B4 ) )
& ~ ( member_nat_set_int @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_253_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_254_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_255_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_256_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_257_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_258_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_259_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_260_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_261_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_262_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_263_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_264_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_265_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_266_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_267_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_268_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_269_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_270_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_271_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_272_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_273_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_274_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_275_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_276_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_277_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_278_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_279_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_280_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_281_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_282_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_283_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_284_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_285_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_286_empty__not__UNIV,axiom,
bot_bot_set_set_int != top_top_set_set_int ).
% empty_not_UNIV
thf(fact_287_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_288_empty__not__UNIV,axiom,
bot_bot_set_int != top_top_set_int ).
% empty_not_UNIV
thf(fact_289_insert__UNIV,axiom,
! [X: set_int] :
( ( insert_set_int @ X @ top_top_set_set_int )
= top_top_set_set_int ) ).
% insert_UNIV
thf(fact_290_insert__UNIV,axiom,
! [X: int] :
( ( insert_int @ X @ top_top_set_int )
= top_top_set_int ) ).
% insert_UNIV
thf(fact_291_singletonD,axiom,
! [B: set_set_int,A: set_set_int] :
( ( member_set_set_int @ B @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_292_singletonD,axiom,
! [B: nat > set_int,A: nat > set_int] :
( ( member_nat_set_int @ B @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_293_singletonD,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_294_singletonD,axiom,
! [B: set_int,A: set_int] :
( ( member_set_int @ B @ ( insert_set_int @ A @ bot_bot_set_set_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_295_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_296_singleton__iff,axiom,
! [B: set_set_int,A: set_set_int] :
( ( member_set_set_int @ B @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_297_singleton__iff,axiom,
! [B: nat > set_int,A: nat > set_int] :
( ( member_nat_set_int @ B @ ( insert_nat_set_int @ A @ bot_bo8417611410066262939et_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_298_singleton__iff,axiom,
! [B: int,A: int] :
( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_299_singleton__iff,axiom,
! [B: set_int,A: set_int] :
( ( member_set_int @ B @ ( insert_set_int @ A @ bot_bot_set_set_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_300_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_301_doubleton__eq__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
= ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_302_doubleton__eq__iff,axiom,
! [A: set_int,B: set_int,C: set_int,D: set_int] :
( ( ( insert_set_int @ A @ ( insert_set_int @ B @ bot_bot_set_set_int ) )
= ( insert_set_int @ C @ ( insert_set_int @ D @ bot_bot_set_set_int ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_303_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_304_insert__not__empty,axiom,
! [A: int,A2: set_int] :
( ( insert_int @ A @ A2 )
!= bot_bot_set_int ) ).
% insert_not_empty
thf(fact_305_insert__not__empty,axiom,
! [A: set_int,A2: set_set_int] :
( ( insert_set_int @ A @ A2 )
!= bot_bot_set_set_int ) ).
% insert_not_empty
thf(fact_306_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_307_singleton__inject,axiom,
! [A: int,B: int] :
( ( ( insert_int @ A @ bot_bot_set_int )
= ( insert_int @ B @ bot_bot_set_int ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_308_singleton__inject,axiom,
! [A: set_int,B: set_int] :
( ( ( insert_set_int @ A @ bot_bot_set_set_int )
= ( insert_set_int @ B @ bot_bot_set_set_int ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_309_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_310_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_311_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_312_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_313_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_314_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_315_int_OIdl__subset__ideal_H,axiom,
! [A: int,B: int] :
( ( member_int @ A @ 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 @ A @ 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 @ A @ ( 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_316_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_317_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_318_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_319_n__ge__0,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% n_ge_0
thf(fact_320_subset__antisym,axiom,
! [A2: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_321_subset__antisym,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ( ( ord_le4403425263959731960et_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_322_subset__antisym,axiom,
! [A2: set_set_set_int,B2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ( ( ord_le4317611570275147438et_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_323_subsetI,axiom,
! [A2: set_nat_set_int,B2: set_nat_set_int] :
( ! [X4: nat > set_int] :
( ( member_nat_set_int @ X4 @ A2 )
=> ( member_nat_set_int @ X4 @ B2 ) )
=> ( ord_le5995675665013768039et_int @ A2 @ B2 ) ) ).
% subsetI
thf(fact_324_subsetI,axiom,
! [A2: set_int,B2: set_int] :
( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( member_int @ X4 @ B2 ) )
=> ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% subsetI
thf(fact_325_subsetI,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ! [X4: set_int] :
( ( member_set_int @ X4 @ A2 )
=> ( member_set_int @ X4 @ B2 ) )
=> ( ord_le4403425263959731960et_int @ A2 @ B2 ) ) ).
% subsetI
thf(fact_326_subsetI,axiom,
! [A2: set_set_set_int,B2: set_set_set_int] :
( ! [X4: set_set_int] :
( ( member_set_set_int @ X4 @ A2 )
=> ( member_set_set_int @ X4 @ B2 ) )
=> ( ord_le4317611570275147438et_int @ A2 @ B2 ) ) ).
% subsetI
thf(fact_327_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_328_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_329_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_330_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_331_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_332_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_333_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_334_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_335_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_336_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_337_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_338_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_339_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_340_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_341_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_342_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_343_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_344_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_345_empty__subsetI,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% empty_subsetI
thf(fact_346_empty__subsetI,axiom,
! [A2: set_set_int] : ( ord_le4403425263959731960et_int @ bot_bot_set_set_int @ A2 ) ).
% empty_subsetI
thf(fact_347_empty__subsetI,axiom,
! [A2: set_set_set_int] : ( ord_le4317611570275147438et_int @ bot_bo2384636101374064866et_int @ A2 ) ).
% empty_subsetI
thf(fact_348_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_349_subset__empty,axiom,
! [A2: set_int] :
( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
= ( A2 = bot_bot_set_int ) ) ).
% subset_empty
thf(fact_350_subset__empty,axiom,
! [A2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ bot_bot_set_set_int )
= ( A2 = bot_bot_set_set_int ) ) ).
% subset_empty
thf(fact_351_subset__empty,axiom,
! [A2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ bot_bo2384636101374064866et_int )
= ( A2 = bot_bo2384636101374064866et_int ) ) ).
% subset_empty
thf(fact_352_insert__subset,axiom,
! [X: nat > set_int,A2: set_nat_set_int,B2: set_nat_set_int] :
( ( ord_le5995675665013768039et_int @ ( insert_nat_set_int @ X @ A2 ) @ B2 )
= ( ( member_nat_set_int @ X @ B2 )
& ( ord_le5995675665013768039et_int @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_353_insert__subset,axiom,
! [X: int,A2: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ ( insert_int @ X @ A2 ) @ B2 )
= ( ( member_int @ X @ B2 )
& ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_354_insert__subset,axiom,
! [X: set_int,A2: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( insert_set_int @ X @ A2 ) @ B2 )
= ( ( member_set_int @ X @ B2 )
& ( ord_le4403425263959731960et_int @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_355_insert__subset,axiom,
! [X: set_set_int,A2: set_set_set_int,B2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ ( insert_set_set_int @ X @ A2 ) @ B2 )
= ( ( member_set_set_int @ X @ B2 )
& ( ord_le4317611570275147438et_int @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_356_dvd__add__triv__right__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_357_dvd__add__triv__right__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_358_dvd__add__triv__left__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_359_dvd__add__triv__left__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_360_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_361_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_362_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_363_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_364_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_365_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_366_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_367_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_368_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_369_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_370_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_371_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_372_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_373_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_374_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_375_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_376_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_377_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_378_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_379_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_380_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_381_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_382_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_383_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_384_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_385_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_386_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_387_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_388_dvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_389_dvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_390_dvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_391_dvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_392_dvd__mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_393_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_394_dvd__add__times__triv__right__iff,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_395_dvd__add__times__triv__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_396_dvd__add__times__triv__left__iff,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
= ( dvd_dvd_int @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_397_dvd__add__times__triv__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_398_unit__prod,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_prod
thf(fact_399_unit__prod,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_prod
thf(fact_400_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_401_singleton__insert__inj__eq_H,axiom,
! [A: int,A2: set_int,B: int] :
( ( ( insert_int @ A @ A2 )
= ( insert_int @ B @ bot_bot_set_int ) )
= ( ( A = B )
& ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_402_singleton__insert__inj__eq_H,axiom,
! [A: set_int,A2: set_set_int,B: set_int] :
( ( ( insert_set_int @ A @ A2 )
= ( insert_set_int @ B @ bot_bot_set_set_int ) )
= ( ( A = B )
& ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_403_singleton__insert__inj__eq_H,axiom,
! [A: set_set_int,A2: set_set_set_int,B: set_set_int] :
( ( ( insert_set_set_int @ A @ A2 )
= ( insert_set_set_int @ B @ bot_bo2384636101374064866et_int ) )
= ( ( A = B )
& ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ B @ bot_bo2384636101374064866et_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_404_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_405_singleton__insert__inj__eq,axiom,
! [B: int,A: int,A2: set_int] :
( ( ( insert_int @ B @ bot_bot_set_int )
= ( insert_int @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_406_singleton__insert__inj__eq,axiom,
! [B: set_int,A: set_int,A2: set_set_int] :
( ( ( insert_set_int @ B @ bot_bot_set_set_int )
= ( insert_set_int @ A @ A2 ) )
= ( ( A = B )
& ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_407_singleton__insert__inj__eq,axiom,
! [B: set_set_int,A: set_set_int,A2: set_set_set_int] :
( ( ( insert_set_set_int @ B @ bot_bo2384636101374064866et_int )
= ( insert_set_set_int @ A @ A2 ) )
= ( ( A = B )
& ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ B @ bot_bo2384636101374064866et_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_408_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_409_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_410_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_411_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_412_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_413_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_414_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_415_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_416_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_417_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_418_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_419_Collect__mono__iff,axiom,
! [P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
= ( ! [X2: int] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_420_Collect__mono__iff,axiom,
! [P: set_int > $o,Q: set_int > $o] :
( ( ord_le4403425263959731960et_int @ ( collect_set_int @ P ) @ ( collect_set_int @ Q ) )
= ( ! [X2: set_int] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_421_Collect__mono__iff,axiom,
! [P: set_set_int > $o,Q: set_set_int > $o] :
( ( ord_le4317611570275147438et_int @ ( collect_set_set_int @ P ) @ ( collect_set_set_int @ Q ) )
= ( ! [X2: set_set_int] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_422_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_423_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_424_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_425_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_426_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_427_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_428_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_429_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_430_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_431_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_432_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_433_set__eq__subset,axiom,
( ( ^ [Y4: set_int,Z2: set_int] : ( Y4 = Z2 ) )
= ( ^ [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
& ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_434_set__eq__subset,axiom,
( ( ^ [Y4: set_set_int,Z2: set_set_int] : ( Y4 = Z2 ) )
= ( ^ [A4: set_set_int,B5: set_set_int] :
( ( ord_le4403425263959731960et_int @ A4 @ B5 )
& ( ord_le4403425263959731960et_int @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_435_set__eq__subset,axiom,
( ( ^ [Y4: set_set_set_int,Z2: set_set_set_int] : ( Y4 = Z2 ) )
= ( ^ [A4: set_set_set_int,B5: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A4 @ B5 )
& ( ord_le4317611570275147438et_int @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_436_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_437_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_438_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_439_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_440_subset__trans,axiom,
! [A2: set_int,B2: set_int,C3: set_int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ C3 )
=> ( ord_less_eq_set_int @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_441_subset__trans,axiom,
! [A2: set_set_int,B2: set_set_int,C3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ( ( ord_le4403425263959731960et_int @ B2 @ C3 )
=> ( ord_le4403425263959731960et_int @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_442_subset__trans,axiom,
! [A2: set_set_set_int,B2: set_set_set_int,C3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ( ( ord_le4317611570275147438et_int @ B2 @ C3 )
=> ( ord_le4317611570275147438et_int @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_443_Collect__mono,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X4: int] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% Collect_mono
thf(fact_444_Collect__mono,axiom,
! [P: set_int > $o,Q: set_int > $o] :
( ! [X4: set_int] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le4403425263959731960et_int @ ( collect_set_int @ P ) @ ( collect_set_int @ Q ) ) ) ).
% Collect_mono
thf(fact_445_Collect__mono,axiom,
! [P: set_set_int > $o,Q: set_set_int > $o] :
( ! [X4: set_set_int] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le4317611570275147438et_int @ ( collect_set_set_int @ P ) @ ( collect_set_set_int @ Q ) ) ) ).
% Collect_mono
thf(fact_446_subset__refl,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% subset_refl
thf(fact_447_subset__refl,axiom,
! [A2: set_set_int] : ( ord_le4403425263959731960et_int @ A2 @ A2 ) ).
% subset_refl
thf(fact_448_subset__refl,axiom,
! [A2: set_set_set_int] : ( ord_le4317611570275147438et_int @ A2 @ A2 ) ).
% subset_refl
thf(fact_449_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_450_subset__iff,axiom,
( ord_le5995675665013768039et_int
= ( ^ [A4: set_nat_set_int,B5: set_nat_set_int] :
! [T2: nat > set_int] :
( ( member_nat_set_int @ T2 @ A4 )
=> ( member_nat_set_int @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_451_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A4: set_int,B5: set_int] :
! [T2: int] :
( ( member_int @ T2 @ A4 )
=> ( member_int @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_452_subset__iff,axiom,
( ord_le4403425263959731960et_int
= ( ^ [A4: set_set_int,B5: set_set_int] :
! [T2: set_int] :
( ( member_set_int @ T2 @ A4 )
=> ( member_set_int @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_453_subset__iff,axiom,
( ord_le4317611570275147438et_int
= ( ^ [A4: set_set_set_int,B5: set_set_set_int] :
! [T2: set_set_int] :
( ( member_set_set_int @ T2 @ A4 )
=> ( member_set_set_int @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_454_equalityD2,axiom,
! [A2: set_int,B2: set_int] :
( ( A2 = B2 )
=> ( ord_less_eq_set_int @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_455_equalityD2,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ( A2 = B2 )
=> ( ord_le4403425263959731960et_int @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_456_equalityD2,axiom,
! [A2: set_set_set_int,B2: set_set_set_int] :
( ( A2 = B2 )
=> ( ord_le4317611570275147438et_int @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_457_equalityD1,axiom,
! [A2: set_int,B2: set_int] :
( ( A2 = B2 )
=> ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_458_equalityD1,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ( A2 = B2 )
=> ( ord_le4403425263959731960et_int @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_459_equalityD1,axiom,
! [A2: set_set_set_int,B2: set_set_set_int] :
( ( A2 = B2 )
=> ( ord_le4317611570275147438et_int @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_460_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_461_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_462_subset__eq,axiom,
( ord_le5995675665013768039et_int
= ( ^ [A4: set_nat_set_int,B5: set_nat_set_int] :
! [X2: nat > set_int] :
( ( member_nat_set_int @ X2 @ A4 )
=> ( member_nat_set_int @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_463_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A4: set_int,B5: set_int] :
! [X2: int] :
( ( member_int @ X2 @ A4 )
=> ( member_int @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_464_subset__eq,axiom,
( ord_le4403425263959731960et_int
= ( ^ [A4: set_set_int,B5: set_set_int] :
! [X2: set_int] :
( ( member_set_int @ X2 @ A4 )
=> ( member_set_int @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_465_subset__eq,axiom,
( ord_le4317611570275147438et_int
= ( ^ [A4: set_set_set_int,B5: set_set_set_int] :
! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A4 )
=> ( member_set_set_int @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_466_equalityE,axiom,
! [A2: set_int,B2: set_int] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
=> ~ ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_467_equalityE,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ( A2 = B2 )
=> ~ ( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ~ ( ord_le4403425263959731960et_int @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_468_equalityE,axiom,
! [A2: set_set_set_int,B2: set_set_set_int] :
( ( A2 = B2 )
=> ~ ( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ~ ( ord_le4317611570275147438et_int @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_469_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_470_subsetD,axiom,
! [A2: set_nat_set_int,B2: set_nat_set_int,C: nat > set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ B2 )
=> ( ( member_nat_set_int @ C @ A2 )
=> ( member_nat_set_int @ C @ B2 ) ) ) ).
% subsetD
thf(fact_471_subsetD,axiom,
! [A2: set_int,B2: set_int,C: int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B2 ) ) ) ).
% subsetD
thf(fact_472_subsetD,axiom,
! [A2: set_set_int,B2: set_set_int,C: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ( ( member_set_int @ C @ A2 )
=> ( member_set_int @ C @ B2 ) ) ) ).
% subsetD
thf(fact_473_subsetD,axiom,
! [A2: set_set_set_int,B2: set_set_set_int,C: set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ( ( member_set_set_int @ C @ A2 )
=> ( member_set_set_int @ C @ B2 ) ) ) ).
% subsetD
thf(fact_474_in__mono,axiom,
! [A2: set_nat_set_int,B2: set_nat_set_int,X: nat > set_int] :
( ( ord_le5995675665013768039et_int @ A2 @ B2 )
=> ( ( member_nat_set_int @ X @ A2 )
=> ( member_nat_set_int @ X @ B2 ) ) ) ).
% in_mono
thf(fact_475_in__mono,axiom,
! [A2: set_int,B2: set_int,X: int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( member_int @ X @ A2 )
=> ( member_int @ X @ B2 ) ) ) ).
% in_mono
thf(fact_476_in__mono,axiom,
! [A2: set_set_int,B2: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ( ( member_set_int @ X @ A2 )
=> ( member_set_int @ X @ B2 ) ) ) ).
% in_mono
thf(fact_477_in__mono,axiom,
! [A2: set_set_set_int,B2: set_set_set_int,X: set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ( ( member_set_set_int @ X @ A2 )
=> ( member_set_set_int @ X @ B2 ) ) ) ).
% in_mono
thf(fact_478_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_479_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_480_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_481_dvd__trans,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ C )
=> ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_trans
thf(fact_482_dvd__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_trans
thf(fact_483_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_484_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_485_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_486_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_487_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_488_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_489_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_490_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_491_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_492_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_493_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_494_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_495_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_496_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_497_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_498_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_499_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_500_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_501_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_502_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_503_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_504_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_505_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_506_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_507_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_508_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_509_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_510_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_511_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_512_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_513_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_514_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_515_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_516_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_517_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_518_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_519_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_520_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_521_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_522_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_523_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_524_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_525_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_526_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_527_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_528_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_529_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_530_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_531_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_532_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_533_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_534_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_535_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_536_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_537_dvd__add__right__iff,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_538_dvd__add__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_add_right_iff
thf(fact_539_dvd__add__left__iff,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_540_dvd__add__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% dvd_add_left_iff
thf(fact_541_dvd__add,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_542_dvd__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% dvd_add
thf(fact_543_dvd__triv__right,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% dvd_triv_right
thf(fact_544_dvd__triv__right,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_545_dvd__mult__right,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ B @ C ) ) ).
% dvd_mult_right
thf(fact_546_dvd__mult__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_547_mult__dvd__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_548_mult__dvd__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_549_dvd__triv__left,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% dvd_triv_left
thf(fact_550_dvd__triv__left,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_551_dvd__mult__left,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ A @ C ) ) ).
% dvd_mult_left
thf(fact_552_dvd__mult__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ).
% dvd_mult_left
thf(fact_553_dvd__mult2,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_554_dvd__mult2,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_555_dvd__mult,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult
thf(fact_556_dvd__mult,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_557_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B3: int,A3: int] :
? [K3: int] :
( A3
= ( times_times_int @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_558_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B3: nat,A3: nat] :
? [K3: nat] :
( A3
= ( times_times_nat @ B3 @ K3 ) ) ) ) ).
% dvd_def
thf(fact_559_dvdI,axiom,
! [A: int,B: int,K: int] :
( ( A
= ( times_times_int @ B @ K ) )
=> ( dvd_dvd_int @ B @ A ) ) ).
% dvdI
thf(fact_560_dvdI,axiom,
! [A: nat,B: nat,K: nat] :
( ( A
= ( times_times_nat @ B @ K ) )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% dvdI
thf(fact_561_dvdE,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ~ ! [K2: int] :
( A
!= ( times_times_int @ B @ K2 ) ) ) ).
% dvdE
thf(fact_562_dvdE,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ~ ! [K2: nat] :
( A
!= ( times_times_nat @ B @ K2 ) ) ) ).
% dvdE
thf(fact_563_dvd__unit__imp__unit,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_564_dvd__unit__imp__unit,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_565_unit__imp__dvd,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_566_unit__imp__dvd,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_567_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_568_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_569_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_570_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_571_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_572_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_573_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_574_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_575_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_576_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_577_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_578_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_579_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_580_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_581_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_582_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_583_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_584_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_585_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_586_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_587_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_588_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_589_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_590_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_591_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_592_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_593_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_594_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C4: nat] :
( B3
= ( plus_plus_nat @ A3 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_595_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_596_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_597_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C5: nat] :
( B
!= ( plus_plus_nat @ A @ C5 ) ) ) ).
% less_eqE
thf(fact_598_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_599_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_600_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_601_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_602_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_603_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_604_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_605_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_606_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_607_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_608_UNIV_I2_J,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ top_top_set_int ) ).
% UNIV(2)
thf(fact_609_UNIV_I2_J,axiom,
! [A2: set_set_int] : ( ord_le4403425263959731960et_int @ A2 @ top_top_set_set_int ) ).
% UNIV(2)
thf(fact_610_UNIV_I2_J,axiom,
! [A2: set_set_set_int] : ( ord_le4317611570275147438et_int @ A2 @ top_to5524576366173240574et_int ) ).
% UNIV(2)
thf(fact_611_subset__UNIV,axiom,
! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ top_top_set_int ) ).
% subset_UNIV
thf(fact_612_subset__UNIV,axiom,
! [A2: set_set_int] : ( ord_le4403425263959731960et_int @ A2 @ top_top_set_set_int ) ).
% subset_UNIV
thf(fact_613_subset__UNIV,axiom,
! [A2: set_set_set_int] : ( ord_le4317611570275147438et_int @ A2 @ top_to5524576366173240574et_int ) ).
% subset_UNIV
thf(fact_614_subset__insertI2,axiom,
! [A2: set_int,B2: set_int,B: int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_615_subset__insertI2,axiom,
! [A2: set_set_int,B2: set_set_int,B: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_616_subset__insertI2,axiom,
! [A2: set_set_set_int,B2: set_set_set_int,B: set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_617_subset__insertI,axiom,
! [B2: set_int,A: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int @ A @ B2 ) ) ).
% subset_insertI
thf(fact_618_subset__insertI,axiom,
! [B2: set_set_int,A: set_int] : ( ord_le4403425263959731960et_int @ B2 @ ( insert_set_int @ A @ B2 ) ) ).
% subset_insertI
thf(fact_619_subset__insertI,axiom,
! [B2: set_set_set_int,A: set_set_int] : ( ord_le4317611570275147438et_int @ B2 @ ( insert_set_set_int @ A @ B2 ) ) ).
% subset_insertI
thf(fact_620_subset__insert,axiom,
! [X: nat > set_int,A2: set_nat_set_int,B2: set_nat_set_int] :
( ~ ( member_nat_set_int @ X @ A2 )
=> ( ( ord_le5995675665013768039et_int @ A2 @ ( insert_nat_set_int @ X @ B2 ) )
= ( ord_le5995675665013768039et_int @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_621_subset__insert,axiom,
! [X: int,A2: set_int,B2: set_int] :
( ~ ( member_int @ X @ A2 )
=> ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) )
= ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_622_subset__insert,axiom,
! [X: set_int,A2: set_set_int,B2: set_set_int] :
( ~ ( member_set_int @ X @ A2 )
=> ( ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X @ B2 ) )
= ( ord_le4403425263959731960et_int @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_623_subset__insert,axiom,
! [X: set_set_int,A2: set_set_set_int,B2: set_set_set_int] :
( ~ ( member_set_set_int @ X @ A2 )
=> ( ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ X @ B2 ) )
= ( ord_le4317611570275147438et_int @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_624_insert__mono,axiom,
! [C3: set_int,D2: set_int,A: int] :
( ( ord_less_eq_set_int @ C3 @ D2 )
=> ( ord_less_eq_set_int @ ( insert_int @ A @ C3 ) @ ( insert_int @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_625_insert__mono,axiom,
! [C3: set_set_int,D2: set_set_int,A: set_int] :
( ( ord_le4403425263959731960et_int @ C3 @ D2 )
=> ( ord_le4403425263959731960et_int @ ( insert_set_int @ A @ C3 ) @ ( insert_set_int @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_626_insert__mono,axiom,
! [C3: set_set_set_int,D2: set_set_set_int,A: set_set_int] :
( ( ord_le4317611570275147438et_int @ C3 @ D2 )
=> ( ord_le4317611570275147438et_int @ ( insert_set_set_int @ A @ C3 ) @ ( insert_set_set_int @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_627_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_628_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_629_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_630_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
= ( dvd_dvd_int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_631_zdvd__period,axiom,
! [A: int,D: int,X: int,T: int,C: int] :
( ( dvd_dvd_int @ A @ D )
=> ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
= ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% zdvd_period
thf(fact_632_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_633_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_634_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_635_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_636_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_637_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_638_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_639_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_640_convex__bound__lt,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X @ A )
=> ( ( ord_less_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_641_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_642_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_643_unit__mult__right__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_644_unit__mult__right__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ B @ A )
= ( times_times_nat @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_645_unit__mult__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_646_unit__mult__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_647_mult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_648_mult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_649_dvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_650_dvd__mult__unit__iff_H,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_651_mult__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_652_mult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_653_dvd__mult__unit__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_654_dvd__mult__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_655_is__unit__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% is_unit_mult_iff
thf(fact_656_is__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% is_unit_mult_iff
thf(fact_657_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_658_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_659_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_660_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_661_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C5: nat] :
( ( B
= ( plus_plus_nat @ A @ C5 ) )
=> ( C5 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_662_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_663_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_664_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_665_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_666_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_667_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_668_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_669_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_670_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_671_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_672_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_673_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_674_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_675_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_676_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_677_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_678_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_679_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_680_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_681_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_682_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_683_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_684_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_685_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_686_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_687_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_688_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_689_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_690_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_691_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_692_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_693_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_694_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_695_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_696_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_697_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_698_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_699_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_700_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_701_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_702_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_703_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_704_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_705_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_706_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_707_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_708_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_709_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_710_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_711_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_712_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_713_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_714_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_715_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_716_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_717_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_718_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_719_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_720_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_721_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_722_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_723_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_724_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_725_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_726_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_727_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_728_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_729_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_730_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_731_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_732_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_733_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_734_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_735_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_736_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_737_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_738_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_739_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_740_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_741_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_742_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_743_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_744_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_745_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_746_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_747_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_748_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_749_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_750_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_751_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_752_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_753_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_754_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_755_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_756_subset__singleton__iff,axiom,
! [X5: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_757_subset__singleton__iff,axiom,
! [X5: set_int,A: int] :
( ( ord_less_eq_set_int @ X5 @ ( insert_int @ A @ bot_bot_set_int ) )
= ( ( X5 = bot_bot_set_int )
| ( X5
= ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% subset_singleton_iff
thf(fact_758_subset__singleton__iff,axiom,
! [X5: set_set_int,A: set_int] :
( ( ord_le4403425263959731960et_int @ X5 @ ( insert_set_int @ A @ bot_bot_set_set_int ) )
= ( ( X5 = bot_bot_set_set_int )
| ( X5
= ( insert_set_int @ A @ bot_bot_set_set_int ) ) ) ) ).
% subset_singleton_iff
thf(fact_759_subset__singleton__iff,axiom,
! [X5: set_set_set_int,A: set_set_int] :
( ( ord_le4317611570275147438et_int @ X5 @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) )
= ( ( X5 = bot_bo2384636101374064866et_int )
| ( X5
= ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) ) ) ).
% subset_singleton_iff
thf(fact_760_subset__singletonD,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_761_subset__singletonD,axiom,
! [A2: set_int,X: int] :
( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) )
=> ( ( A2 = bot_bot_set_int )
| ( A2
= ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% subset_singletonD
thf(fact_762_subset__singletonD,axiom,
! [A2: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X @ bot_bot_set_set_int ) )
=> ( ( A2 = bot_bot_set_set_int )
| ( A2
= ( insert_set_int @ X @ bot_bot_set_set_int ) ) ) ) ).
% subset_singletonD
thf(fact_763_subset__singletonD,axiom,
! [A2: set_set_set_int,X: set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ X @ bot_bo2384636101374064866et_int ) )
=> ( ( A2 = bot_bo2384636101374064866et_int )
| ( A2
= ( insert_set_set_int @ X @ bot_bo2384636101374064866et_int ) ) ) ) ).
% subset_singletonD
thf(fact_764_unit__dvdE,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [C5: int] :
( B
!= ( times_times_int @ A @ C5 ) ) ) ) ).
% unit_dvdE
thf(fact_765_unit__dvdE,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [C5: nat] :
( B
!= ( times_times_nat @ A @ C5 ) ) ) ) ).
% unit_dvdE
thf(fact_766_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_767_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_768_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_769_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_770_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_771_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_772_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_773_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_774_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_775_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_776_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_777_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_778_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_779_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_780_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_781_int_Ocgenideal__minimal,axiom,
! [J2: set_int,A: int] :
( ( ideal_6787631597145370931t_unit @ J2 @ ( 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 ) ) ) )
=> ( ( member_int @ A @ J2 )
=> ( ord_less_eq_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 ) ) ) @ A ) @ J2 ) ) ) ).
% int.cgenideal_minimal
thf(fact_782_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_783_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_784_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_785_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_786_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_787_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_788_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_789_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_790_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_791_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_792_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_793_int_Osubset__Idl__subset,axiom,
! [I2: set_int,H: set_int] :
( ( ord_less_eq_set_int @ I2 @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ H @ I2 )
=> ( 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 ) ) ) @ I2 ) ) ) ) ).
% int.subset_Idl_subset
thf(fact_794_int_Ogenideal__self,axiom,
! [S2: set_int] :
( ( ord_less_eq_set_int @ S2 @ top_top_set_int )
=> ( ord_less_eq_set_int @ S2 @ ( 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 ) ) ) @ S2 ) ) ) ).
% int.genideal_self
thf(fact_795_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_796_int_Ocgenideal__ideal,axiom,
! [A: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ideal_6787631597145370931t_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 ) ) ) @ A ) @ ( 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_ideal
thf(fact_797_int_Ogenideal__minimal,axiom,
! [I2: set_int,S2: set_int] :
( ( ideal_6787631597145370931t_unit @ I2 @ ( 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 ) ) ) )
=> ( ( ord_less_eq_set_int @ S2 @ I2 )
=> ( 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 ) ) ) @ S2 ) @ I2 ) ) ) ).
% int.genideal_minimal
thf(fact_798_int_OIdl__subset__ideal,axiom,
! [I2: set_int,H: set_int] :
( ( ideal_6787631597145370931t_unit @ I2 @ ( 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 ) ) ) )
=> ( ( ord_less_eq_set_int @ H @ 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 ) ) ) @ H ) @ I2 )
= ( ord_less_eq_set_int @ H @ I2 ) ) ) ) ).
% int.Idl_subset_ideal
thf(fact_799_int_Ogenideal__ideal,axiom,
! [S2: set_int] :
( ( ord_less_eq_set_int @ S2 @ top_top_set_int )
=> ( ideal_6787631597145370931t_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 ) ) ) @ S2 ) @ ( 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.genideal_ideal
thf(fact_800_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_801_int_Omaximalideal__prime,axiom,
! [I2: set_int] :
( ( maxima7040249999675607092t_unit @ I2 @ ( 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 @ I2 @ ( 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_802_unity__coeff__ex,axiom,
! [P: int > $o,L: int] :
( ( ? [X2: int] : ( P @ ( times_times_int @ L @ X2 ) ) )
= ( ? [X2: int] :
( ( dvd_dvd_int @ L @ ( plus_plus_int @ X2 @ zero_zero_int ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_803_unity__coeff__ex,axiom,
! [P: nat > $o,L: nat] :
( ( ? [X2: nat] : ( P @ ( times_times_nat @ L @ X2 ) ) )
= ( ? [X2: nat] :
( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
& ( P @ X2 ) ) ) ) ).
% unity_coeff_ex
thf(fact_804_mult__le__cancel__iff2,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_805_mult__le__cancel__iff1,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_806_psubsetI,axiom,
! [A2: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_int @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_807_psubsetI,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_set_int @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_808_psubsetI,axiom,
! [A2: set_set_set_int,B2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le4562804192517611682et_int @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_809_int_Ole__refl,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ord_less_eq_int @ X @ X ) ) ).
% int.le_refl
thf(fact_810_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_811_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_812_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_813_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_814_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_815_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_816_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_817_int__dvd__int__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ).
% int_dvd_int_iff
thf(fact_818_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_819_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_820_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_821_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_822_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_823_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_824_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_825_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_826_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_827_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P2: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_828_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P2: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_829_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_830_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_831_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_832_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_833_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_834_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X7: int] :
( ( P @ X7 )
=> ( P @ ( plus_plus_int @ X7 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_835_psubsetE,axiom,
! [A2: set_int,B2: set_int] :
( ( ord_less_set_int @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_836_psubsetE,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B2 )
=> ~ ( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ( ord_le4403425263959731960et_int @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_837_psubsetE,axiom,
! [A2: set_set_set_int,B2: set_set_set_int] :
( ( ord_le4562804192517611682et_int @ A2 @ B2 )
=> ~ ( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ( ord_le4317611570275147438et_int @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_838_psubset__eq,axiom,
( ord_less_set_int
= ( ^ [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_839_psubset__eq,axiom,
( ord_less_set_set_int
= ( ^ [A4: set_set_int,B5: set_set_int] :
( ( ord_le4403425263959731960et_int @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_840_psubset__eq,axiom,
( ord_le4562804192517611682et_int
= ( ^ [A4: set_set_set_int,B5: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_841_psubset__imp__subset,axiom,
! [A2: set_int,B2: set_int] :
( ( ord_less_set_int @ A2 @ B2 )
=> ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_842_psubset__imp__subset,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B2 )
=> ( ord_le4403425263959731960et_int @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_843_psubset__imp__subset,axiom,
! [A2: set_set_set_int,B2: set_set_set_int] :
( ( ord_le4562804192517611682et_int @ A2 @ B2 )
=> ( ord_le4317611570275147438et_int @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_844_psubset__subset__trans,axiom,
! [A2: set_int,B2: set_int,C3: set_int] :
( ( ord_less_set_int @ A2 @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ C3 )
=> ( ord_less_set_int @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_845_psubset__subset__trans,axiom,
! [A2: set_set_int,B2: set_set_int,C3: set_set_int] :
( ( ord_less_set_set_int @ A2 @ B2 )
=> ( ( ord_le4403425263959731960et_int @ B2 @ C3 )
=> ( ord_less_set_set_int @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_846_psubset__subset__trans,axiom,
! [A2: set_set_set_int,B2: set_set_set_int,C3: set_set_set_int] :
( ( ord_le4562804192517611682et_int @ A2 @ B2 )
=> ( ( ord_le4317611570275147438et_int @ B2 @ C3 )
=> ( ord_le4562804192517611682et_int @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_847_subset__not__subset__eq,axiom,
( ord_less_set_int
= ( ^ [A4: set_int,B5: set_int] :
( ( ord_less_eq_set_int @ A4 @ B5 )
& ~ ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_848_subset__not__subset__eq,axiom,
( ord_less_set_set_int
= ( ^ [A4: set_set_int,B5: set_set_int] :
( ( ord_le4403425263959731960et_int @ A4 @ B5 )
& ~ ( ord_le4403425263959731960et_int @ B5 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_849_subset__not__subset__eq,axiom,
( ord_le4562804192517611682et_int
= ( ^ [A4: set_set_set_int,B5: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A4 @ B5 )
& ~ ( ord_le4317611570275147438et_int @ B5 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_850_subset__psubset__trans,axiom,
! [A2: set_int,B2: set_int,C3: set_int] :
( ( ord_less_eq_set_int @ A2 @ B2 )
=> ( ( ord_less_set_int @ B2 @ C3 )
=> ( ord_less_set_int @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_851_subset__psubset__trans,axiom,
! [A2: set_set_int,B2: set_set_int,C3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B2 )
=> ( ( ord_less_set_set_int @ B2 @ C3 )
=> ( ord_less_set_set_int @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_852_subset__psubset__trans,axiom,
! [A2: set_set_set_int,B2: set_set_set_int,C3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B2 )
=> ( ( ord_le4562804192517611682et_int @ B2 @ C3 )
=> ( ord_le4562804192517611682et_int @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_853_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A4: set_int,B5: set_int] :
( ( ord_less_set_int @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_854_subset__iff__psubset__eq,axiom,
( ord_le4403425263959731960et_int
= ( ^ [A4: set_set_int,B5: set_set_int] :
( ( ord_less_set_set_int @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_855_subset__iff__psubset__eq,axiom,
( ord_le4317611570275147438et_int
= ( ^ [A4: set_set_set_int,B5: set_set_set_int] :
( ( ord_le4562804192517611682et_int @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_856_not__psubset__empty,axiom,
! [A2: set_int] :
~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% not_psubset_empty
thf(fact_857_not__psubset__empty,axiom,
! [A2: set_set_int] :
~ ( ord_less_set_set_int @ A2 @ bot_bot_set_set_int ) ).
% not_psubset_empty
thf(fact_858_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_859_int_Olless__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C @ top_top_set_int )
=> ( ord_less_int @ A @ C ) ) ) ) ) ) ).
% int.lless_trans
thf(fact_860_int_Olless__antisym,axiom,
! [A: int,B: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ) ) ).
% int.lless_antisym
thf(fact_861_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_862_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_863_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_864_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_865_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_866_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_867_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_868_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_869_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_870_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_871_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_872_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_873_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_874_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_875_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_876_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_877_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_878_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_879_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_880_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_881_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_882_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_883_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_884_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_885_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_886_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_887_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_888_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_889_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_890_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_891_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_892_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_893_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_894_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_895_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_896_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z3: int] :
? [N3: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_897_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_898_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_899_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_900_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_901_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_902_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_903_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_904_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N )
=> ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_905_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M @ N )
=> ( ( dvd_dvd_int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_906_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_907_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_908_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_909_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z5 @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P2 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_910_pinf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ Z5 @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P2 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_911_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z5 @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P2 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_912_pinf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ Z5 @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P2 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_913_pinf_I3_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z5 @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_914_pinf_I3_J,axiom,
! [T: int] :
? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ Z5 @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_915_pinf_I4_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z5 @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_916_pinf_I4_J,axiom,
! [T: int] :
? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ Z5 @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_917_pinf_I5_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z5 @ X7 )
=> ~ ( ord_less_nat @ X7 @ T ) ) ).
% pinf(5)
thf(fact_918_pinf_I5_J,axiom,
! [T: int] :
? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ Z5 @ X7 )
=> ~ ( ord_less_int @ X7 @ T ) ) ).
% pinf(5)
thf(fact_919_pinf_I7_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z5 @ X7 )
=> ( ord_less_nat @ T @ X7 ) ) ).
% pinf(7)
thf(fact_920_pinf_I7_J,axiom,
! [T: int] :
? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ Z5 @ X7 )
=> ( ord_less_int @ T @ X7 ) ) ).
% pinf(7)
thf(fact_921_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z5 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P2 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_922_minf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z5 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P2 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_923_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z5 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P2 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_924_minf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P2 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z5 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P2 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_925_minf_I3_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z5 )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_926_minf_I3_J,axiom,
! [T: int] :
? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z5 )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_927_minf_I4_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z5 )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_928_minf_I4_J,axiom,
! [T: int] :
? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z5 )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_929_minf_I5_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z5 )
=> ( ord_less_nat @ X7 @ T ) ) ).
% minf(5)
thf(fact_930_minf_I5_J,axiom,
! [T: int] :
? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z5 )
=> ( ord_less_int @ X7 @ T ) ) ).
% minf(5)
thf(fact_931_minf_I7_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z5 )
=> ~ ( ord_less_nat @ T @ X7 ) ) ).
% minf(7)
thf(fact_932_minf_I7_J,axiom,
! [T: int] :
? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z5 )
=> ~ ( ord_less_int @ T @ X7 ) ) ).
% minf(7)
thf(fact_933_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_934_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_935_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_936_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_937_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_938_pinf_I6_J,axiom,
! [T: int] :
? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ Z5 @ X7 )
=> ~ ( ord_less_eq_int @ X7 @ T ) ) ).
% pinf(6)
thf(fact_939_pinf_I6_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z5 @ X7 )
=> ~ ( ord_less_eq_nat @ X7 @ T ) ) ).
% pinf(6)
thf(fact_940_pinf_I8_J,axiom,
! [T: int] :
? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ Z5 @ X7 )
=> ( ord_less_eq_int @ T @ X7 ) ) ).
% pinf(8)
thf(fact_941_pinf_I8_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z5 @ X7 )
=> ( ord_less_eq_nat @ T @ X7 ) ) ).
% pinf(8)
thf(fact_942_minf_I6_J,axiom,
! [T: int] :
? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z5 )
=> ( ord_less_eq_int @ X7 @ T ) ) ).
% minf(6)
thf(fact_943_minf_I6_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z5 )
=> ( ord_less_eq_nat @ X7 @ T ) ) ).
% minf(6)
thf(fact_944_minf_I8_J,axiom,
! [T: int] :
? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z5 )
=> ~ ( ord_less_eq_int @ T @ X7 ) ) ).
% minf(8)
thf(fact_945_minf_I8_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z5 )
=> ~ ( ord_less_eq_nat @ T @ X7 ) ) ).
% minf(8)
thf(fact_946_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_947_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_948_mult__less__iff1,axiom,
! [Z: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_949_minf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z5 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X7 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X7 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_950_minf_I10_J,axiom,
! [D: int,S: int] :
? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z5 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ S ) ) ) ) ) ).
% minf(10)
thf(fact_951_minf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z5 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X7 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X7 @ S ) ) ) ) ).
% minf(9)
thf(fact_952_minf_I9_J,axiom,
! [D: int,S: int] :
? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z5 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ S ) ) ) ) ).
% minf(9)
thf(fact_953_pinf_I10_J,axiom,
! [D: nat,S: nat] :
? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z5 @ X7 )
=> ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X7 @ S ) ) )
= ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X7 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_954_pinf_I10_J,axiom,
! [D: int,S: int] :
? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ Z5 @ X7 )
=> ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ S ) ) )
= ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ S ) ) ) ) ) ).
% pinf(10)
thf(fact_955_pinf_I9_J,axiom,
! [D: nat,S: nat] :
? [Z5: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z5 @ X7 )
=> ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X7 @ S ) )
= ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X7 @ S ) ) ) ) ).
% pinf(9)
thf(fact_956_pinf_I9_J,axiom,
! [D: int,S: int] :
? [Z5: int] :
! [X7: int] :
( ( ord_less_int @ Z5 @ X7 )
=> ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ S ) )
= ( dvd_dvd_int @ D @ ( plus_plus_int @ X7 @ S ) ) ) ) ).
% pinf(9)
thf(fact_957_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_958_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_959_zdvd__mono,axiom,
! [K: int,M: int,T: int] :
( ( K != zero_zero_int )
=> ( ( dvd_dvd_int @ M @ T )
= ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% zdvd_mono
thf(fact_960_prime__primeideal,axiom,
! [P3: int] :
( ( factor1798656936486255268me_int @ P3 )
=> ( 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 @ P3 @ 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_961_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_962_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_963_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_964_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_965_psubsetD,axiom,
! [A2: set_int,B2: set_int,C: int] :
( ( ord_less_set_int @ A2 @ B2 )
=> ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_966_psubsetD,axiom,
! [A2: set_set_int,B2: set_set_int,C: set_int] :
( ( ord_less_set_set_int @ A2 @ B2 )
=> ( ( member_set_int @ C @ A2 )
=> ( member_set_int @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_967_psubsetD,axiom,
! [A2: set_set_set_int,B2: set_set_set_int,C: set_set_int] :
( ( ord_le4562804192517611682et_int @ A2 @ B2 )
=> ( ( member_set_set_int @ C @ A2 )
=> ( member_set_set_int @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_968_psubsetD,axiom,
! [A2: set_nat_set_int,B2: set_nat_set_int,C: nat > set_int] :
( ( ord_le2931775347370382171et_int @ A2 @ B2 )
=> ( ( member_nat_set_int @ C @ A2 )
=> ( member_nat_set_int @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_969_int_Olless__eq,axiom,
( ord_less_int
= ( ^ [X2: int,Y6: int] :
( ( ord_less_eq_int @ X2 @ Y6 )
& ( X2 != Y6 ) ) ) ) ).
% int.lless_eq
thf(fact_970_field_Oaxioms_I1_J,axiom,
! [R2: partia2818514838349642498t_unit] :
( ( field_5117527561578272769t_unit @ R2 )
=> ( domain1430183510194609567t_unit @ R2 ) ) ).
% field.axioms(1)
thf(fact_971_field_Oaxioms_I1_J,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( field_5943785737635511755t_unit @ R2 )
=> ( domain6183376680155302761t_unit @ R2 ) ) ).
% field.axioms(1)
thf(fact_972_field_Oaxioms_I1_J,axiom,
! [R2: partia3601206958761670294t_unit] :
( ( field_4969826516285655701t_unit @ R2 )
=> ( domain2593739924244766771t_unit @ R2 ) ) ).
% field.axioms(1)
thf(fact_973_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_974_int_Ocring__fieldI2,axiom,
( ( zero_zero_int != one_one_int )
=> ( ! [A5: int] :
( ( member_int @ A5 @ top_top_set_int )
=> ( ( A5 != zero_zero_int )
=> ? [X7: int] :
( ( member_int @ X7 @ top_top_set_int )
& ( ( times_times_int @ A5 @ X7 )
= 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_975_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_976_prime__int__altdef,axiom,
( factor1798656936486255268me_int
= ( ^ [P4: int] :
( ( ord_less_int @ one_one_int @ P4 )
& ! [M3: int] :
( ( ord_less_eq_int @ zero_zero_int @ M3 )
=> ( ( dvd_dvd_int @ M3 @ P4 )
=> ( ( M3 = one_one_int )
| ( M3 = P4 ) ) ) ) ) ) ) ).
% prime_int_altdef
thf(fact_977_Primes_Oprime__int__iff,axiom,
( factor1798656936486255268me_int
= ( ^ [N3: int] :
( ( ord_less_int @ one_one_int @ N3 )
& ! [M3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ M3 )
& ( dvd_dvd_int @ M3 @ N3 ) )
=> ( ( M3 = one_one_int )
| ( M3 = N3 ) ) ) ) ) ) ).
% Primes.prime_int_iff
thf(fact_978_prime__divisors__induct,axiom,
! [P: int > $o,X: int] :
( ( P @ zero_zero_int )
=> ( ! [X4: int] :
( ( dvd_dvd_int @ X4 @ one_one_int )
=> ( P @ X4 ) )
=> ( ! [P5: int,X4: int] :
( ( factor1798656936486255268me_int @ P5 )
=> ( ( P @ X4 )
=> ( P @ ( times_times_int @ P5 @ X4 ) ) ) )
=> ( P @ X ) ) ) ) ).
% prime_divisors_induct
thf(fact_979_prime__divisors__induct,axiom,
! [P: nat > $o,X: nat] :
( ( P @ zero_zero_nat )
=> ( ! [X4: nat] :
( ( dvd_dvd_nat @ X4 @ one_one_nat )
=> ( P @ X4 ) )
=> ( ! [P5: nat,X4: nat] :
( ( factor1801147406995305544me_nat @ P5 )
=> ( ( P @ X4 )
=> ( P @ ( times_times_nat @ P5 @ X4 ) ) ) )
=> ( P @ X ) ) ) ) ).
% prime_divisors_induct
thf(fact_980_prime__nat__int__transfer,axiom,
! [N: nat] :
( ( factor1798656936486255268me_int @ ( semiri1314217659103216013at_int @ N ) )
= ( factor1801147406995305544me_nat @ N ) ) ).
% prime_nat_int_transfer
thf(fact_981_prime__dvd__mult__eq__nat,axiom,
! [P3: nat,A: nat,B: nat] :
( ( factor1801147406995305544me_nat @ P3 )
=> ( ( dvd_dvd_nat @ P3 @ ( times_times_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ P3 @ A )
| ( dvd_dvd_nat @ P3 @ B ) ) ) ) ).
% prime_dvd_mult_eq_nat
thf(fact_982_prime__factor__nat,axiom,
! [N: nat] :
( ( N != one_one_nat )
=> ? [P5: nat] :
( ( factor1801147406995305544me_nat @ P5 )
& ( dvd_dvd_nat @ P5 @ N ) ) ) ).
% prime_factor_nat
thf(fact_983_prime__ge__1__nat,axiom,
! [P3: nat] :
( ( factor1801147406995305544me_nat @ P3 )
=> ( ord_less_eq_nat @ one_one_nat @ P3 ) ) ).
% prime_ge_1_nat
thf(fact_984_bigger__prime,axiom,
! [N: nat] :
? [P5: nat] :
( ( factor1801147406995305544me_nat @ P5 )
& ( ord_less_nat @ N @ P5 ) ) ).
% bigger_prime
thf(fact_985_prime__gt__0__nat,axiom,
! [P3: nat] :
( ( factor1801147406995305544me_nat @ P3 )
=> ( ord_less_nat @ zero_zero_nat @ P3 ) ) ).
% prime_gt_0_nat
thf(fact_986_prime__gt__1__nat,axiom,
! [P3: nat] :
( ( factor1801147406995305544me_nat @ P3 )
=> ( ord_less_nat @ one_one_nat @ P3 ) ) ).
% prime_gt_1_nat
thf(fact_987_prime__product,axiom,
! [P3: nat,Q3: nat] :
( ( factor1801147406995305544me_nat @ ( times_times_nat @ P3 @ Q3 ) )
=> ( ( P3 = one_one_nat )
| ( Q3 = one_one_nat ) ) ) ).
% prime_product
thf(fact_988_not__prime__eq__prod__nat,axiom,
! [M: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ~ ( factor1801147406995305544me_nat @ M )
=> ? [N2: nat,K2: nat] :
( ( N2
= ( times_times_nat @ M @ K2 ) )
& ( ord_less_nat @ one_one_nat @ M )
& ( ord_less_nat @ M @ N2 )
& ( ord_less_nat @ one_one_nat @ K2 )
& ( ord_less_nat @ K2 @ N2 ) ) ) ) ).
% not_prime_eq_prod_nat
thf(fact_989_prime__nat__iff,axiom,
( factor1801147406995305544me_nat
= ( ^ [N3: nat] :
( ( ord_less_nat @ one_one_nat @ N3 )
& ! [M3: nat] :
( ( dvd_dvd_nat @ M3 @ N3 )
=> ( ( M3 = one_one_nat )
| ( M3 = N3 ) ) ) ) ) ) ).
% prime_nat_iff
thf(fact_990_prime__nat__not__dvd,axiom,
! [P3: nat,N: nat] :
( ( factor1801147406995305544me_nat @ P3 )
=> ( ( ord_less_nat @ N @ P3 )
=> ( ( N != one_one_nat )
=> ~ ( dvd_dvd_nat @ N @ P3 ) ) ) ) ).
% prime_nat_not_dvd
thf(fact_991_not__prime__0,axiom,
~ ( factor1798656936486255268me_int @ zero_zero_int ) ).
% not_prime_0
thf(fact_992_not__prime__0,axiom,
~ ( factor1801147406995305544me_nat @ zero_zero_nat ) ).
% not_prime_0
thf(fact_993_not__prime__1,axiom,
~ ( factor1798656936486255268me_int @ one_one_int ) ).
% not_prime_1
thf(fact_994_not__prime__1,axiom,
~ ( factor1801147406995305544me_nat @ one_one_nat ) ).
% not_prime_1
thf(fact_995_primes__dvd__imp__eq,axiom,
! [P3: int,Q3: int] :
( ( factor1798656936486255268me_int @ P3 )
=> ( ( factor1798656936486255268me_int @ Q3 )
=> ( ( dvd_dvd_int @ P3 @ Q3 )
=> ( P3 = Q3 ) ) ) ) ).
% primes_dvd_imp_eq
thf(fact_996_primes__dvd__imp__eq,axiom,
! [P3: nat,Q3: nat] :
( ( factor1801147406995305544me_nat @ P3 )
=> ( ( factor1801147406995305544me_nat @ Q3 )
=> ( ( dvd_dvd_nat @ P3 @ Q3 )
=> ( P3 = Q3 ) ) ) ) ).
% primes_dvd_imp_eq
thf(fact_997_prime__dvd__mult__iff,axiom,
! [P3: int,A: int,B: int] :
( ( factor1798656936486255268me_int @ P3 )
=> ( ( dvd_dvd_int @ P3 @ ( times_times_int @ A @ B ) )
= ( ( dvd_dvd_int @ P3 @ A )
| ( dvd_dvd_int @ P3 @ B ) ) ) ) ).
% prime_dvd_mult_iff
thf(fact_998_prime__dvd__mult__iff,axiom,
! [P3: nat,A: nat,B: nat] :
( ( factor1801147406995305544me_nat @ P3 )
=> ( ( dvd_dvd_nat @ P3 @ ( times_times_nat @ A @ B ) )
= ( ( dvd_dvd_nat @ P3 @ A )
| ( dvd_dvd_nat @ P3 @ B ) ) ) ) ).
% prime_dvd_mult_iff
thf(fact_999_prime__dvd__multD,axiom,
! [P3: int,A: int,B: int] :
( ( factor1798656936486255268me_int @ P3 )
=> ( ( dvd_dvd_int @ P3 @ ( times_times_int @ A @ B ) )
=> ( ( dvd_dvd_int @ P3 @ A )
| ( dvd_dvd_int @ P3 @ B ) ) ) ) ).
% prime_dvd_multD
thf(fact_1000_prime__dvd__multD,axiom,
! [P3: nat,A: nat,B: nat] :
( ( factor1801147406995305544me_nat @ P3 )
=> ( ( dvd_dvd_nat @ P3 @ ( times_times_nat @ A @ B ) )
=> ( ( dvd_dvd_nat @ P3 @ A )
| ( dvd_dvd_nat @ P3 @ B ) ) ) ) ).
% prime_dvd_multD
thf(fact_1001_not__prime__unit,axiom,
! [X: int] :
( ( dvd_dvd_int @ X @ one_one_int )
=> ~ ( factor1798656936486255268me_int @ X ) ) ).
% not_prime_unit
thf(fact_1002_not__prime__unit,axiom,
! [X: nat] :
( ( dvd_dvd_nat @ X @ one_one_nat )
=> ~ ( factor1801147406995305544me_nat @ X ) ) ).
% not_prime_unit
thf(fact_1003_prime__ge__0__int,axiom,
! [P3: int] :
( ( factor1798656936486255268me_int @ P3 )
=> ( ord_less_eq_int @ zero_zero_int @ P3 ) ) ).
% prime_ge_0_int
thf(fact_1004_prime__ge__1__int,axiom,
! [P3: int] :
( ( factor1798656936486255268me_int @ P3 )
=> ( ord_less_eq_int @ one_one_int @ P3 ) ) ).
% prime_ge_1_int
thf(fact_1005_prime__gt__0__int,axiom,
! [P3: int] :
( ( factor1798656936486255268me_int @ P3 )
=> ( ord_less_int @ zero_zero_int @ P3 ) ) ).
% prime_gt_0_int
thf(fact_1006_prime__gt__1__int,axiom,
! [P3: int] :
( ( factor1798656936486255268me_int @ P3 )
=> ( ord_less_int @ one_one_int @ P3 ) ) ).
% prime_gt_1_int
thf(fact_1007_prime__dvd__mult__eq__int,axiom,
! [P3: int,A: int,B: int] :
( ( factor1798656936486255268me_int @ P3 )
=> ( ( dvd_dvd_int @ P3 @ ( times_times_int @ A @ B ) )
= ( ( dvd_dvd_int @ P3 @ A )
| ( dvd_dvd_int @ P3 @ B ) ) ) ) ).
% prime_dvd_mult_eq_int
thf(fact_1008_prime__divisorE,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ~ ( dvd_dvd_int @ A @ one_one_int )
=> ~ ! [P5: int] :
( ( factor1798656936486255268me_int @ P5 )
=> ~ ( dvd_dvd_int @ P5 @ A ) ) ) ) ).
% prime_divisorE
thf(fact_1009_prime__divisorE,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ~ ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ! [P5: nat] :
( ( factor1801147406995305544me_nat @ P5 )
=> ~ ( dvd_dvd_nat @ P5 @ A ) ) ) ) ).
% prime_divisorE
thf(fact_1010_prime__divisor__exists,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ~ ( dvd_dvd_int @ A @ one_one_int )
=> ? [B6: int] :
( ( dvd_dvd_int @ B6 @ A )
& ( factor1798656936486255268me_int @ B6 ) ) ) ) ).
% prime_divisor_exists
thf(fact_1011_prime__divisor__exists,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ~ ( dvd_dvd_nat @ A @ one_one_nat )
=> ? [B6: nat] :
( ( dvd_dvd_nat @ B6 @ A )
& ( factor1801147406995305544me_nat @ B6 ) ) ) ) ).
% prime_divisor_exists
thf(fact_1012_prime__int__not__dvd,axiom,
! [P3: int,N: int] :
( ( factor1798656936486255268me_int @ P3 )
=> ( ( ord_less_int @ N @ P3 )
=> ( ( ord_less_int @ one_one_int @ N )
=> ~ ( dvd_dvd_int @ N @ P3 ) ) ) ) ).
% prime_int_not_dvd
thf(fact_1013_bezout__add__strong__nat,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ? [D3: nat,X4: nat,Y3: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( times_times_nat @ A @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1014_dvd__nat__bounds,axiom,
! [P3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ P3 )
=> ( ( dvd_dvd_nat @ N @ P3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
& ( ord_less_eq_nat @ N @ P3 ) ) ) ) ).
% dvd_nat_bounds
thf(fact_1015_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
thf(fact_1016_gcd__nat_Oasym,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ~ ( ( dvd_dvd_nat @ B @ A )
& ( B != A ) ) ) ).
% gcd_nat.asym
thf(fact_1017_gcd__nat_Orefl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% gcd_nat.refl
thf(fact_1018_gcd__nat_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% gcd_nat.trans
thf(fact_1019_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( dvd_dvd_nat @ A3 @ B3 )
& ( dvd_dvd_nat @ B3 @ A3 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_1020_gcd__nat_Oirrefl,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ A @ A )
& ( A != A ) ) ).
% gcd_nat.irrefl
thf(fact_1021_gcd__nat_Oantisym,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( A = B ) ) ) ).
% gcd_nat.antisym
thf(fact_1022_gcd__nat_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_1023_gcd__nat_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_1024_gcd__nat_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_1025_gcd__nat_Ostrict__iff__not,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_1026_gcd__nat_Oorder__iff__strict,axiom,
( dvd_dvd_nat
= ( ^ [A3: nat,B3: nat] :
( ( ( dvd_dvd_nat @ A3 @ B3 )
& ( A3 != B3 ) )
| ( A3 = B3 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_1027_gcd__nat_Ostrict__iff__order,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_1028_gcd__nat_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( dvd_dvd_nat @ A @ B ) ) ).
% gcd_nat.strict_implies_order
thf(fact_1029_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( A != B ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_1030_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_1031_division__decomp,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
=> ? [B7: int,C6: int] :
( ( A
= ( times_times_int @ B7 @ C6 ) )
& ( dvd_dvd_int @ B7 @ B )
& ( dvd_dvd_int @ C6 @ C ) ) ) ).
% division_decomp
thf(fact_1032_division__decomp,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
=> ? [B7: nat,C6: nat] :
( ( A
= ( times_times_nat @ B7 @ C6 ) )
& ( dvd_dvd_nat @ B7 @ B )
& ( dvd_dvd_nat @ C6 @ C ) ) ) ).
% division_decomp
thf(fact_1033_dvd__productE,axiom,
! [P3: int,A: int,B: int] :
( ( dvd_dvd_int @ P3 @ ( times_times_int @ A @ B ) )
=> ~ ! [X4: int,Y3: int] :
( ( P3
= ( times_times_int @ X4 @ Y3 ) )
=> ( ( dvd_dvd_int @ X4 @ A )
=> ~ ( dvd_dvd_int @ Y3 @ B ) ) ) ) ).
% dvd_productE
thf(fact_1034_dvd__productE,axiom,
! [P3: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ P3 @ ( times_times_nat @ A @ B ) )
=> ~ ! [X4: nat,Y3: nat] :
( ( P3
= ( times_times_nat @ X4 @ Y3 ) )
=> ( ( dvd_dvd_nat @ X4 @ A )
=> ~ ( dvd_dvd_nat @ Y3 @ B ) ) ) ) ).
% dvd_productE
thf(fact_1035_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B6: nat] :
( ( P @ A5 @ B6 )
= ( P @ B6 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B6: nat] :
( ( P @ A5 @ B6 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B6 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1036_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_1037_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_1038_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_1039_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_1040_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1041_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_1042_bezout__add__nat,axiom,
! [A: nat,B: nat] :
? [D3: nat,X4: nat,Y3: nat] :
( ( dvd_dvd_nat @ D3 @ A )
& ( dvd_dvd_nat @ D3 @ B )
& ( ( ( times_times_nat @ A @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) )
| ( ( times_times_nat @ B @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D3 ) ) ) ) ).
% bezout_add_nat
thf(fact_1043_bezout__lemma__nat,axiom,
! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
( ( dvd_dvd_nat @ D @ A )
=> ( ( dvd_dvd_nat @ D @ B )
=> ( ( ( ( times_times_nat @ A @ X )
= ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
| ( ( times_times_nat @ B @ X )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
=> ? [X4: nat,Y3: nat] :
( ( dvd_dvd_nat @ D @ A )
& ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
& ( ( ( times_times_nat @ A @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D ) )
| ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X4 )
= ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1044_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_1045_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_1046_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_1047_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_1048_zfact__prime__is__field,axiom,
! [P3: nat] :
( ( factor1801147406995305544me_nat @ P3 )
=> ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ P3 ) ) ) ) ).
% zfact_prime_is_field
thf(fact_1049_ZFact__prime__is__domain,axiom,
! [P3: int] :
( ( factor1798656936486255268me_int @ P3 )
=> ( domain6183376680155302761t_unit @ ( zFact @ P3 ) ) ) ).
% ZFact_prime_is_domain
thf(fact_1050_r_Osemiring__axioms,axiom,
semiri8708897239777792527t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.semiring_axioms
thf(fact_1051_primeideal_Oquotient__is__domain,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( primei2109666362732673920t_unit @ I2 @ R2 )
=> ( domain6183376680155302761t_unit @ ( factRi5755170488246124606t_unit @ R2 @ I2 ) ) ) ).
% primeideal.quotient_is_domain
thf(fact_1052_primeideal_Oquotient__is__domain,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( primei350866878734230858t_unit @ I2 @ R2 )
=> ( domain2593739924244766771t_unit @ ( factRi3149420076008518152t_unit @ R2 @ I2 ) ) ) ).
% primeideal.quotient_is_domain
thf(fact_1053_r_Oabelian__monoid__axioms,axiom,
abelia3815030880812984441t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.abelian_monoid_axioms
thf(fact_1054_semiring_Oaxioms_I1_J,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( abelia3815030880812984441t_unit @ R2 ) ) ).
% semiring.axioms(1)
thf(fact_1055_semiring_Oaxioms_I1_J,axiom,
! [R2: partia2818514838349642498t_unit] :
( ( semiri1037594922888297541t_unit @ R2 )
=> ( abelia6702305049305627311t_unit @ R2 ) ) ).
% semiring.axioms(1)
thf(fact_1056_r_Omaximalideal__prime,axiom,
! [I2: set_set_int] :
( ( maxima6262477034536100350t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( primei350866878734230858t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.maximalideal_prime
thf(fact_1057_r_Odomain__eq__zeroprimeideal,axiom,
( ( domain6183376680155302761t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( primei350866878734230858t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.domain_eq_zeroprimeideal
thf(fact_1058_r_Ozeroprimeideal__domainI,axiom,
( ( primei350866878734230858t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( domain6183376680155302761t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.zeroprimeideal_domainI
thf(fact_1059_r_Ozeromaximalideal__eq__field,axiom,
( ( maxima6262477034536100350t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.zeromaximalideal_eq_field
thf(fact_1060_r_Ozeromaximalideal__fieldI,axiom,
( ( maxima6262477034536100350t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.zeromaximalideal_fieldI
thf(fact_1061_ring__record__simps_I11_J,axiom,
! [Carrier2: set_set_int,Mult2: set_int > set_int > set_int,One2: set_int,Zero2: set_int,Add2: set_int > set_int > set_int,More: product_unit] :
( ( zero_s6269048424454532197t_unit @ ( partia768359127423289238t_unit @ Carrier2 @ ( monoid3816432266719977132t_unit @ Mult2 @ One2 @ ( ring_e3077432647605977236t_unit @ Zero2 @ Add2 @ More ) ) ) )
= Zero2 ) ).
% ring_record_simps(11)
thf(fact_1062_ring__record__simps_I11_J,axiom,
! [Carrier2: set_int,Mult2: int > int > int,One2: int,Zero2: int,Add2: int > int > int,More: product_unit] :
( ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ Carrier2 @ ( monoid6080699973261426200t_unit @ Mult2 @ One2 @ ( ring_e5272872978682396362t_unit @ Zero2 @ Add2 @ More ) ) ) )
= Zero2 ) ).
% ring_record_simps(11)
thf(fact_1063_int_Oabelian__monoid__axioms,axiom,
abelia6702305049305627311t_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.abelian_monoid_axioms
thf(fact_1064_int_Osemiring__axioms,axiom,
semiri1037594922888297541t_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.semiring_axioms
thf(fact_1065_r_OFactRing__zeroideal_I2_J,axiom,
is_rin4964436081320486288t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) ).
% r.FactRing_zeroideal(2)
thf(fact_1066_r_OFactRing__zeroideal_I1_J,axiom,
is_rin1729965262454186108t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.FactRing_zeroideal(1)
thf(fact_1067_r_Ozeropideal,axiom,
princi8860937869964495385t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.zeropideal
thf(fact_1068_int__zero__eq,axiom,
( ( zero_i2266321264637750939t_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_zero_eq
thf(fact_1069_r_Ogenideal__zero,axiom,
( ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) ).
% r.genideal_zero
thf(fact_1070_field_Ozeromaximalideal,axiom,
! [R2: partia2818514838349642498t_unit] :
( ( field_5117527561578272769t_unit @ R2 )
=> ( maxima7040249999675607092t_unit @ ( insert_int @ ( zero_i2266321264637750939t_unit @ R2 ) @ bot_bot_set_int ) @ R2 ) ) ).
% field.zeromaximalideal
thf(fact_1071_field_Ozeromaximalideal,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( field_5943785737635511755t_unit @ R2 )
=> ( maxima6262477034536100350t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) @ R2 ) ) ).
% field.zeromaximalideal
thf(fact_1072_domain_Ozeroprimeideal,axiom,
! [R2: partia3601206958761670294t_unit] :
( ( domain2593739924244766771t_unit @ R2 )
=> ( primei8868987738197308948t_unit @ ( insert_set_set_int @ ( zero_s2946895028292161839t_unit @ R2 ) @ bot_bo2384636101374064866et_int ) @ R2 ) ) ).
% domain.zeroprimeideal
thf(fact_1073_domain_Ozeroprimeideal,axiom,
! [R2: partia2818514838349642498t_unit] :
( ( domain1430183510194609567t_unit @ R2 )
=> ( primei2109666362732673920t_unit @ ( insert_int @ ( zero_i2266321264637750939t_unit @ R2 ) @ bot_bot_set_int ) @ R2 ) ) ).
% domain.zeroprimeideal
thf(fact_1074_domain_Ozeroprimeideal,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( domain6183376680155302761t_unit @ R2 )
=> ( primei350866878734230858t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) @ R2 ) ) ).
% domain.zeroprimeideal
thf(fact_1075_ideal_Ois__ideal,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( ideal_6787631597145370931t_unit @ I2 @ R2 )
=> ( ideal_6787631597145370931t_unit @ I2 @ R2 ) ) ).
% ideal.is_ideal
thf(fact_1076_ideal_Ois__ideal,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
=> ( ideal_7262958097527394045t_unit @ I2 @ R2 ) ) ).
% ideal.is_ideal
thf(fact_1077_ideal_Ois__ideal,axiom,
! [I2: set_set_set_int,R2: partia3601206958761670294t_unit] :
( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
=> ( ideal_1992573625310827463t_unit @ I2 @ R2 ) ) ).
% ideal.is_ideal
thf(fact_1078_primeideal_Oprimeideal,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( primei2109666362732673920t_unit @ I2 @ R2 )
=> ( primei2109666362732673920t_unit @ I2 @ R2 ) ) ).
% primeideal.primeideal
thf(fact_1079_primeideal_Oprimeideal,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( primei350866878734230858t_unit @ I2 @ R2 )
=> ( primei350866878734230858t_unit @ I2 @ R2 ) ) ).
% primeideal.primeideal
thf(fact_1080_maximalideal_Ois__maximalideal,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( maxima7040249999675607092t_unit @ I2 @ R2 )
=> ( maxima7040249999675607092t_unit @ I2 @ R2 ) ) ).
% maximalideal.is_maximalideal
thf(fact_1081_maximalideal_Ois__maximalideal,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( maxima6262477034536100350t_unit @ I2 @ R2 )
=> ( maxima6262477034536100350t_unit @ I2 @ R2 ) ) ).
% maximalideal.is_maximalideal
thf(fact_1082_principalideal_Ois__principalideal,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( princi1768892856804252751t_unit @ I2 @ R2 )
=> ( princi1768892856804252751t_unit @ I2 @ R2 ) ) ).
% principalideal.is_principalideal
thf(fact_1083_principalideal_Ois__principalideal,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( princi8860937869964495385t_unit @ I2 @ R2 )
=> ( princi8860937869964495385t_unit @ I2 @ R2 ) ) ).
% principalideal.is_principalideal
thf(fact_1084_primeideal_Oaxioms_I1_J,axiom,
! [I2: set_set_set_int,R2: partia3601206958761670294t_unit] :
( ( primei8868987738197308948t_unit @ I2 @ R2 )
=> ( ideal_1992573625310827463t_unit @ I2 @ R2 ) ) ).
% primeideal.axioms(1)
thf(fact_1085_primeideal_Oaxioms_I1_J,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( primei2109666362732673920t_unit @ I2 @ R2 )
=> ( ideal_6787631597145370931t_unit @ I2 @ R2 ) ) ).
% primeideal.axioms(1)
thf(fact_1086_primeideal_Oaxioms_I1_J,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( primei350866878734230858t_unit @ I2 @ R2 )
=> ( ideal_7262958097527394045t_unit @ I2 @ R2 ) ) ).
% primeideal.axioms(1)
thf(fact_1087_maximalideal_Oaxioms_I1_J,axiom,
! [I2: set_set_set_int,R2: partia3601206958761670294t_unit] :
( ( maxima8339209307728405192t_unit @ I2 @ R2 )
=> ( ideal_1992573625310827463t_unit @ I2 @ R2 ) ) ).
% maximalideal.axioms(1)
thf(fact_1088_maximalideal_Oaxioms_I1_J,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( maxima7040249999675607092t_unit @ I2 @ R2 )
=> ( ideal_6787631597145370931t_unit @ I2 @ R2 ) ) ).
% maximalideal.axioms(1)
thf(fact_1089_maximalideal_Oaxioms_I1_J,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( maxima6262477034536100350t_unit @ I2 @ R2 )
=> ( ideal_7262958097527394045t_unit @ I2 @ R2 ) ) ).
% maximalideal.axioms(1)
thf(fact_1090_principalideal_Oaxioms_I1_J,axiom,
! [I2: set_set_set_int,R2: partia3601206958761670294t_unit] :
( ( princi840266688866541283t_unit @ I2 @ R2 )
=> ( ideal_1992573625310827463t_unit @ I2 @ R2 ) ) ).
% principalideal.axioms(1)
thf(fact_1091_principalideal_Oaxioms_I1_J,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( princi1768892856804252751t_unit @ I2 @ R2 )
=> ( ideal_6787631597145370931t_unit @ I2 @ R2 ) ) ).
% principalideal.axioms(1)
thf(fact_1092_principalideal_Oaxioms_I1_J,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( princi8860937869964495385t_unit @ I2 @ R2 )
=> ( ideal_7262958097527394045t_unit @ I2 @ R2 ) ) ).
% principalideal.axioms(1)
thf(fact_1093_r_OIdl__subset__ideal_H,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) )
= ( member_set_int @ A @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ) ) ).
% r.Idl_subset_ideal'
thf(fact_1094_r_Ogenideal__self_H,axiom,
! [I: set_int] :
( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ I @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ I @ bot_bot_set_set_int ) ) ) ) ).
% r.genideal_self'
thf(fact_1095_r_Oonepideal,axiom,
princi8860937869964495385t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.onepideal
thf(fact_1096_r_Ooneideal,axiom,
ideal_7262958097527394045t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.oneideal
thf(fact_1097_r_Ocarrier__not__empty,axiom,
( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= bot_bot_set_set_int ) ).
% r.carrier_not_empty
thf(fact_1098_r_Ogenideal__minimal,axiom,
! [I2: set_set_int,S2: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ord_le4403425263959731960et_int @ S2 @ I2 )
=> ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S2 ) @ I2 ) ) ) ).
% r.genideal_minimal
thf(fact_1099_r_Oquot__domain__imp__primeideal,axiom,
! [P: set_set_int] :
( ( ideal_7262958097527394045t_unit @ P @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( domain2593739924244766771t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ P ) )
=> ( primei350866878734230858t_unit @ P @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.quot_domain_imp_primeideal
thf(fact_1100_r_Oquot__domain__iff__primeideal,axiom,
! [P: set_set_int] :
( ( ideal_7262958097527394045t_unit @ P @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( domain2593739924244766771t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ P ) )
= ( primei350866878734230858t_unit @ P @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.quot_domain_iff_primeideal
thf(fact_1101_r_Ozeroideal,axiom,
ideal_7262958097527394045t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.zeroideal
thf(fact_1102_r_OIdl__subset__ideal,axiom,
! [I2: set_set_int,H: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H ) @ I2 )
= ( ord_le4403425263959731960et_int @ H @ I2 ) ) ) ) ).
% r.Idl_subset_ideal
thf(fact_1103_r_Ogenideal__ideal,axiom,
! [S2: set_set_int] :
( ( ord_le4403425263959731960et_int @ S2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ideal_7262958097527394045t_unit @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S2 ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.genideal_ideal
thf(fact_1104_r_Osubset__Idl__subset,axiom,
! [I2: set_set_int,H: set_set_int] :
( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ H @ I2 )
=> ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H ) @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 ) ) ) ) ).
% r.subset_Idl_subset
thf(fact_1105_r_Ogenideal__self,axiom,
! [S2: set_set_int] :
( ( ord_le4403425263959731960et_int @ S2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ S2 @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S2 ) ) ) ).
% r.genideal_self
thf(fact_1106_r_Ozero__closed,axiom,
member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% r.zero_closed
thf(fact_1107_ideal_OIcarr,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit,I: set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
=> ( ( member_set_int @ I @ I2 )
=> ( member_set_int @ I @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ).
% ideal.Icarr
thf(fact_1108_ideal_OIcarr,axiom,
! [I2: set_set_set_int,R2: partia3601206958761670294t_unit,I: set_set_int] :
( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
=> ( ( member_set_set_int @ I @ I2 )
=> ( member_set_set_int @ I @ ( partia1981568256555877781t_unit @ R2 ) ) ) ) ).
% ideal.Icarr
thf(fact_1109_ideal_OIcarr,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit,I: int] :
( ( ideal_6787631597145370931t_unit @ I2 @ R2 )
=> ( ( member_int @ I @ I2 )
=> ( member_int @ I @ ( partia8426541738980984321t_unit @ R2 ) ) ) ) ).
% ideal.Icarr
thf(fact_1110_maximalideal_OI__notcarr,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( maxima6262477034536100350t_unit @ I2 @ R2 )
=> ( ( partia966996272515721803t_unit @ R2 )
!= I2 ) ) ).
% maximalideal.I_notcarr
thf(fact_1111_maximalideal_OI__notcarr,axiom,
! [I2: set_set_set_int,R2: partia3601206958761670294t_unit] :
( ( maxima8339209307728405192t_unit @ I2 @ R2 )
=> ( ( partia1981568256555877781t_unit @ R2 )
!= I2 ) ) ).
% maximalideal.I_notcarr
thf(fact_1112_maximalideal_OI__notcarr,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( maxima7040249999675607092t_unit @ I2 @ R2 )
=> ( ( partia8426541738980984321t_unit @ R2 )
!= I2 ) ) ).
% maximalideal.I_notcarr
thf(fact_1113_primeideal_OI__notcarr,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( primei350866878734230858t_unit @ I2 @ R2 )
=> ( ( partia966996272515721803t_unit @ R2 )
!= I2 ) ) ).
% primeideal.I_notcarr
thf(fact_1114_primeideal_OI__notcarr,axiom,
! [I2: set_set_set_int,R2: partia3601206958761670294t_unit] :
( ( primei8868987738197308948t_unit @ I2 @ R2 )
=> ( ( partia1981568256555877781t_unit @ R2 )
!= I2 ) ) ).
% primeideal.I_notcarr
thf(fact_1115_primeideal_OI__notcarr,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( primei2109666362732673920t_unit @ I2 @ R2 )
=> ( ( partia8426541738980984321t_unit @ R2 )
!= I2 ) ) ).
% primeideal.I_notcarr
thf(fact_1116_abelian__monoidE_I2_J,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( abelia3815030880812984441t_unit @ R2 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% abelian_monoidE(2)
thf(fact_1117_abelian__monoidE_I2_J,axiom,
! [R2: partia3601206958761670294t_unit] :
( ( abelia4259588778567990595t_unit @ R2 )
=> ( member_set_set_int @ ( zero_s2946895028292161839t_unit @ R2 ) @ ( partia1981568256555877781t_unit @ R2 ) ) ) ).
% abelian_monoidE(2)
thf(fact_1118_abelian__monoidE_I2_J,axiom,
! [R2: partia2818514838349642498t_unit] :
( ( abelia6702305049305627311t_unit @ R2 )
=> ( member_int @ ( zero_i2266321264637750939t_unit @ R2 ) @ ( partia8426541738980984321t_unit @ R2 ) ) ) ).
% abelian_monoidE(2)
thf(fact_1119_abelian__monoid_Ozero__closed,axiom,
! [G: partia4934656038542163276t_unit] :
( ( abelia3815030880812984441t_unit @ G )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ G ) @ ( partia966996272515721803t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_1120_abelian__monoid_Ozero__closed,axiom,
! [G: partia3601206958761670294t_unit] :
( ( abelia4259588778567990595t_unit @ G )
=> ( member_set_set_int @ ( zero_s2946895028292161839t_unit @ G ) @ ( partia1981568256555877781t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_1121_abelian__monoid_Ozero__closed,axiom,
! [G: partia2818514838349642498t_unit] :
( ( abelia6702305049305627311t_unit @ G )
=> ( member_int @ ( zero_i2266321264637750939t_unit @ G ) @ ( partia8426541738980984321t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_1122_semiring_Osemiring__simprules_I2_J,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( semiri8708897239777792527t_unit @ R2 )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_1123_semiring_Osemiring__simprules_I2_J,axiom,
! [R2: partia3601206958761670294t_unit] :
( ( semiri9009874074179768025t_unit @ R2 )
=> ( member_set_set_int @ ( zero_s2946895028292161839t_unit @ R2 ) @ ( partia1981568256555877781t_unit @ R2 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_1124_semiring_Osemiring__simprules_I2_J,axiom,
! [R2: partia2818514838349642498t_unit] :
( ( semiri1037594922888297541t_unit @ R2 )
=> ( member_int @ ( zero_i2266321264637750939t_unit @ R2 ) @ ( partia8426541738980984321t_unit @ R2 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_1125_maximalideal_OI__maximal,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit,J2: set_set_int] :
( ( maxima6262477034536100350t_unit @ I2 @ R2 )
=> ( ( ideal_7262958097527394045t_unit @ J2 @ R2 )
=> ( ( ord_le4403425263959731960et_int @ I2 @ J2 )
=> ( ( ord_le4403425263959731960et_int @ J2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( J2 = I2 )
| ( J2
= ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ) ) ).
% maximalideal.I_maximal
thf(fact_1126_maximalideal_OI__maximal,axiom,
! [I2: set_set_set_int,R2: partia3601206958761670294t_unit,J2: set_set_set_int] :
( ( maxima8339209307728405192t_unit @ I2 @ R2 )
=> ( ( ideal_1992573625310827463t_unit @ J2 @ R2 )
=> ( ( ord_le4317611570275147438et_int @ I2 @ J2 )
=> ( ( ord_le4317611570275147438et_int @ J2 @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( J2 = I2 )
| ( J2
= ( partia1981568256555877781t_unit @ R2 ) ) ) ) ) ) ) ).
% maximalideal.I_maximal
thf(fact_1127_maximalideal_OI__maximal,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit,J2: set_int] :
( ( maxima7040249999675607092t_unit @ I2 @ R2 )
=> ( ( ideal_6787631597145370931t_unit @ J2 @ R2 )
=> ( ( ord_less_eq_set_int @ I2 @ J2 )
=> ( ( ord_less_eq_set_int @ J2 @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( J2 = I2 )
| ( J2
= ( partia8426541738980984321t_unit @ R2 ) ) ) ) ) ) ) ).
% maximalideal.I_maximal
thf(fact_1128_maximalidealI,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
=> ( ( ( partia966996272515721803t_unit @ R2 )
!= I2 )
=> ( ! [J4: set_set_int] :
( ( ideal_7262958097527394045t_unit @ J4 @ R2 )
=> ( ( ord_le4403425263959731960et_int @ I2 @ J4 )
=> ( ( ord_le4403425263959731960et_int @ J4 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( J4 = I2 )
| ( J4
= ( partia966996272515721803t_unit @ R2 ) ) ) ) ) )
=> ( maxima6262477034536100350t_unit @ I2 @ R2 ) ) ) ) ).
% maximalidealI
thf(fact_1129_maximalidealI,axiom,
! [I2: set_set_set_int,R2: partia3601206958761670294t_unit] :
( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
=> ( ( ( partia1981568256555877781t_unit @ R2 )
!= I2 )
=> ( ! [J4: set_set_set_int] :
( ( ideal_1992573625310827463t_unit @ J4 @ R2 )
=> ( ( ord_le4317611570275147438et_int @ I2 @ J4 )
=> ( ( ord_le4317611570275147438et_int @ J4 @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( J4 = I2 )
| ( J4
= ( partia1981568256555877781t_unit @ R2 ) ) ) ) ) )
=> ( maxima8339209307728405192t_unit @ I2 @ R2 ) ) ) ) ).
% maximalidealI
thf(fact_1130_maximalidealI,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( ideal_6787631597145370931t_unit @ I2 @ R2 )
=> ( ( ( partia8426541738980984321t_unit @ R2 )
!= I2 )
=> ( ! [J4: set_int] :
( ( ideal_6787631597145370931t_unit @ J4 @ R2 )
=> ( ( ord_less_eq_set_int @ I2 @ J4 )
=> ( ( ord_less_eq_set_int @ J4 @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( J4 = I2 )
| ( J4
= ( partia8426541738980984321t_unit @ R2 ) ) ) ) ) )
=> ( maxima7040249999675607092t_unit @ I2 @ R2 ) ) ) ) ).
% maximalidealI
thf(fact_1131_principalideal_Ogenerate,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( princi8860937869964495385t_unit @ I2 @ R2 )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ R2 ) )
& ( I2
= ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ X4 @ bot_bot_set_set_int ) ) ) ) ) ).
% principalideal.generate
thf(fact_1132_principalideal_Ogenerate,axiom,
! [I2: set_set_set_int,R2: partia3601206958761670294t_unit] :
( ( princi840266688866541283t_unit @ I2 @ R2 )
=> ? [X4: set_set_int] :
( ( member_set_set_int @ X4 @ ( partia1981568256555877781t_unit @ R2 ) )
& ( I2
= ( genide1354039622117027493t_unit @ R2 @ ( insert_set_set_int @ X4 @ bot_bo2384636101374064866et_int ) ) ) ) ) ).
% principalideal.generate
thf(fact_1133_principalideal_Ogenerate,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( princi1768892856804252751t_unit @ I2 @ R2 )
=> ? [X4: int] :
( ( member_int @ X4 @ ( partia8426541738980984321t_unit @ R2 ) )
& ( I2
= ( genide1613390280493775889t_unit @ R2 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ) ).
% principalideal.generate
thf(fact_1134_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_1135_principalidealI,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
=> ( ? [X7: set_int] :
( ( member_set_int @ X7 @ ( partia966996272515721803t_unit @ R2 ) )
& ( I2
= ( genide1545711809618862555t_unit @ R2 @ ( insert_set_int @ X7 @ bot_bot_set_set_int ) ) ) )
=> ( princi8860937869964495385t_unit @ I2 @ R2 ) ) ) ).
% principalidealI
thf(fact_1136_principalidealI,axiom,
! [I2: set_set_set_int,R2: partia3601206958761670294t_unit] :
( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
=> ( ? [X7: set_set_int] :
( ( member_set_set_int @ X7 @ ( partia1981568256555877781t_unit @ R2 ) )
& ( I2
= ( genide1354039622117027493t_unit @ R2 @ ( insert_set_set_int @ X7 @ bot_bo2384636101374064866et_int ) ) ) )
=> ( princi840266688866541283t_unit @ I2 @ R2 ) ) ) ).
% principalidealI
thf(fact_1137_principalidealI,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( ideal_6787631597145370931t_unit @ I2 @ R2 )
=> ( ? [X7: int] :
( ( member_int @ X7 @ ( partia8426541738980984321t_unit @ R2 ) )
& ( I2
= ( genide1613390280493775889t_unit @ R2 @ ( insert_int @ X7 @ bot_bot_set_int ) ) ) )
=> ( princi1768892856804252751t_unit @ I2 @ R2 ) ) ) ).
% principalidealI
thf(fact_1138_r_OboundD__carrier,axiom,
! [N: nat,F: nat > set_int,M: nat] :
( ( bound_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_set_int @ ( F @ M ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.boundD_carrier
thf(fact_1139_r_Ocgenideal__eq__genideal,axiom,
! [I: set_int] :
( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I )
= ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ I @ bot_bot_set_set_int ) ) ) ) ).
% r.cgenideal_eq_genideal
thf(fact_1140_r_Oa__lcos__mult__one,axiom,
! [M5: set_set_int] :
( ( ord_le4403425263959731960et_int @ M5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ M5 )
= M5 ) ) ).
% r.a_lcos_mult_one
thf(fact_1141_r_Ocgenideal__self,axiom,
! [I: set_int] :
( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ I @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I ) ) ) ).
% r.cgenideal_self
thf(fact_1142_r_Oa__l__coset__subset__G,axiom,
! [H: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ H ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.a_l_coset_subset_G
thf(fact_1143_r_Ocgenideal__ideal,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ideal_7262958097527394045t_unit @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.cgenideal_ideal
thf(fact_1144_r_Ocgenideal__minimal,axiom,
! [J2: set_set_int,A: set_int] :
( ( ideal_7262958097527394045t_unit @ J2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ A @ J2 )
=> ( ord_le4403425263959731960et_int @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A ) @ J2 ) ) ) ).
% r.cgenideal_minimal
thf(fact_1145_r_Ocgenideal__is__principalideal,axiom,
! [I: set_int] :
( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( princi8860937869964495385t_unit @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.cgenideal_is_principalideal
thf(fact_1146_r_Oquot__carr,axiom,
! [I2: set_set_int,Y: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_set_int @ Y @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 ) ) )
=> ( ord_le4403425263959731960et_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.quot_carr
thf(fact_1147_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_1148_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_1149_r_Oideal__is__subalgebra,axiom,
! [K4: set_set_int,I2: set_set_int] :
( ( ord_le4403425263959731960et_int @ K4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( embedd2743979684206749024t_unit @ K4 @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.ideal_is_subalgebra
thf(fact_1150_r_Oset__add__zero,axiom,
! [A2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) @ A2 )
= A2 ) ) ).
% r.set_add_zero
thf(fact_1151_r_Ocarrier__is__subalgebra,axiom,
! [K4: set_set_int] :
( ( ord_le4403425263959731960et_int @ K4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( embedd2743979684206749024t_unit @ K4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.carrier_is_subalgebra
thf(fact_1152_r_Oadd__ideals,axiom,
! [I2: set_set_int,J2: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ideal_7262958097527394045t_unit @ J2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ideal_7262958097527394045t_unit @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ J2 ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.add_ideals
thf(fact_1153_r_Oset__add__closed,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A2 @ B2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.set_add_closed
thf(fact_1154_r_Oset__add__comm,axiom,
! [I2: set_set_int,J2: set_set_int] :
( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ J2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ J2 )
= ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ J2 @ I2 ) ) ) ) ).
% r.set_add_comm
thf(fact_1155_r_Osetadd__subset__G,axiom,
! [H: set_set_int,K4: set_set_int] :
( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H @ K4 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.setadd_subset_G
thf(fact_1156_r_Osubalgebra__in__carrier,axiom,
! [K4: set_set_int,V2: set_set_int] :
( ( embedd2743979684206749024t_unit @ K4 @ V2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ord_le4403425263959731960et_int @ V2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.subalgebra_in_carrier
thf(fact_1157_r_Ocanonical__proj__vimage__in__carrier,axiom,
! [I2: set_set_int,J2: set_set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ord_le4317611570275147438et_int @ J2 @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 ) ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ J2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.canonical_proj_vimage_in_carrier
thf(fact_1158_r_Oadd__additive__subgroups,axiom,
! [H: set_set_int,K4: set_set_int] :
( ( additi7073586575563672860t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( additi7073586575563672860t_unit @ K4 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( additi7073586575563672860t_unit @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H @ K4 ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.add_additive_subgroups
thf(fact_1159_r_Obound__upD,axiom,
! [F: nat > set_int] :
( ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [N2: nat] : ( bound_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ N2 @ F ) ) ).
% r.bound_upD
thf(fact_1160_r_Oquot__ideal__imp__ring__ideal,axiom,
! [I2: set_set_int,J2: set_set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ideal_1992573625310827463t_unit @ J2 @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 ) )
=> ( ideal_7262958097527394045t_unit @ ( comple7281953568134767595et_int @ J2 ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.quot_ideal_imp_ring_ideal
thf(fact_1161_ideal_Oaxioms_I1_J,axiom,
! [I2: set_set_set_int,R2: partia3601206958761670294t_unit] :
( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
=> ( additi2938837234532341222t_unit @ I2 @ R2 ) ) ).
% ideal.axioms(1)
thf(fact_1162_ideal_Oaxioms_I1_J,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
=> ( additi7073586575563672860t_unit @ I2 @ R2 ) ) ).
% ideal.axioms(1)
thf(fact_1163_ideal_Oaxioms_I1_J,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( ideal_6787631597145370931t_unit @ I2 @ R2 )
=> ( additi1413919225584036690t_unit @ I2 @ R2 ) ) ).
% ideal.axioms(1)
thf(fact_1164_ccpo__Sup__singleton,axiom,
! [X: set_set_int] :
( ( comple7281953568134767595et_int @ ( insert_set_set_int @ X @ bot_bo2384636101374064866et_int ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_1165_ccpo__Sup__singleton,axiom,
! [X: set_int] :
( ( comple3221217463730067765et_int @ ( insert_set_int @ X @ bot_bot_set_set_int ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_1166_cSup__singleton,axiom,
! [X: int] :
( ( complete_Sup_Sup_int @ ( insert_int @ X @ bot_bot_set_int ) )
= X ) ).
% cSup_singleton
thf(fact_1167_cSup__singleton,axiom,
! [X: set_set_int] :
( ( comple7281953568134767595et_int @ ( insert_set_set_int @ X @ bot_bo2384636101374064866et_int ) )
= X ) ).
% cSup_singleton
thf(fact_1168_cSup__singleton,axiom,
! [X: nat] :
( ( complete_Sup_Sup_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
= X ) ).
% cSup_singleton
thf(fact_1169_cSup__singleton,axiom,
! [X: set_int] :
( ( comple3221217463730067765et_int @ ( insert_set_int @ X @ bot_bot_set_set_int ) )
= X ) ).
% cSup_singleton
thf(fact_1170_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_1171_Sup__empty,axiom,
( ( comple7281953568134767595et_int @ bot_bo2384636101374064866et_int )
= bot_bot_set_set_int ) ).
% Sup_empty
thf(fact_1172_Sup__empty,axiom,
( ( comple3221217463730067765et_int @ bot_bot_set_set_int )
= bot_bot_set_int ) ).
% Sup_empty
thf(fact_1173_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1174_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1175_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_set_int] :
( ( ( comple7281953568134767595et_int @ A2 )
= bot_bot_set_set_int )
= ( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_int ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1176_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_int] :
( ( ( comple3221217463730067765et_int @ A2 )
= bot_bot_set_int )
= ( ! [X2: set_int] :
( ( member_set_int @ X2 @ A2 )
=> ( X2 = bot_bot_set_int ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_1177_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( X2 = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1178_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_set_int] :
( ( bot_bot_set_set_int
= ( comple7281953568134767595et_int @ A2 ) )
= ( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_int ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1179_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_int] :
( ( bot_bot_set_int
= ( comple3221217463730067765et_int @ A2 ) )
= ( ! [X2: set_int] :
( ( member_set_int @ X2 @ A2 )
=> ( X2 = bot_bot_set_int ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_1180_Sup__UNIV,axiom,
( ( comple7281953568134767595et_int @ top_to5524576366173240574et_int )
= top_top_set_set_int ) ).
% Sup_UNIV
thf(fact_1181_Sup__UNIV,axiom,
( ( comple3221217463730067765et_int @ top_top_set_set_int )
= top_top_set_int ) ).
% Sup_UNIV
thf(fact_1182_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1183_bot__set__def,axiom,
( bot_bot_set_int
= ( collect_int @ bot_bot_int_o ) ) ).
% bot_set_def
thf(fact_1184_bot__set__def,axiom,
( bot_bot_set_set_int
= ( collect_set_int @ bot_bot_set_int_o ) ) ).
% bot_set_def
thf(fact_1185_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1186_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C5: int] :
( ( ord_less_eq_int @ A @ C5 )
& ( ord_less_eq_int @ C5 @ B )
& ! [X7: int] :
( ( ( ord_less_eq_int @ A @ X7 )
& ( ord_less_int @ X7 @ C5 ) )
=> ( P @ X7 ) )
& ! [D4: int] :
( ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ D4 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_int @ D4 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1187_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C5: nat] :
( ( ord_less_eq_nat @ A @ C5 )
& ( ord_less_eq_nat @ C5 @ B )
& ! [X7: nat] :
( ( ( ord_less_eq_nat @ A @ X7 )
& ( ord_less_nat @ X7 @ C5 ) )
=> ( P @ X7 ) )
& ! [D4: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D4 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D4 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1188_Sup__eqI,axiom,
! [A2: set_nat_set_int,X: nat > set_int] :
( ! [Y3: nat > set_int] :
( ( member_nat_set_int @ Y3 @ A2 )
=> ( ord_le3704955753469811889et_int @ Y3 @ X ) )
=> ( ! [Y3: nat > set_int] :
( ! [Z4: nat > set_int] :
( ( member_nat_set_int @ Z4 @ A2 )
=> ( ord_le3704955753469811889et_int @ Z4 @ Y3 ) )
=> ( ord_le3704955753469811889et_int @ X @ Y3 ) )
=> ( ( comple93655007136625956et_int @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_1189_Sup__eqI,axiom,
! [A2: set_set_set_set_int,X: set_set_set_int] :
( ! [Y3: set_set_set_int] :
( ( member7356822600254261989et_int @ Y3 @ A2 )
=> ( ord_le4317611570275147438et_int @ Y3 @ X ) )
=> ( ! [Y3: set_set_set_int] :
( ! [Z4: set_set_set_int] :
( ( member7356822600254261989et_int @ Z4 @ A2 )
=> ( ord_le4317611570275147438et_int @ Z4 @ Y3 ) )
=> ( ord_le4317611570275147438et_int @ X @ Y3 ) )
=> ( ( comple1756060948637632417et_int @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_1190_Sup__eqI,axiom,
! [A2: set_set_set_int,X: set_set_int] :
( ! [Y3: set_set_int] :
( ( member_set_set_int @ Y3 @ A2 )
=> ( ord_le4403425263959731960et_int @ Y3 @ X ) )
=> ( ! [Y3: set_set_int] :
( ! [Z4: set_set_int] :
( ( member_set_set_int @ Z4 @ A2 )
=> ( ord_le4403425263959731960et_int @ Z4 @ Y3 ) )
=> ( ord_le4403425263959731960et_int @ X @ Y3 ) )
=> ( ( comple7281953568134767595et_int @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_1191_Sup__eqI,axiom,
! [A2: set_set_int,X: set_int] :
( ! [Y3: set_int] :
( ( member_set_int @ Y3 @ A2 )
=> ( ord_less_eq_set_int @ Y3 @ X ) )
=> ( ! [Y3: set_int] :
( ! [Z4: set_int] :
( ( member_set_int @ Z4 @ A2 )
=> ( ord_less_eq_set_int @ Z4 @ Y3 ) )
=> ( ord_less_eq_set_int @ X @ Y3 ) )
=> ( ( comple3221217463730067765et_int @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_1192_Sup__mono,axiom,
! [A2: set_nat_set_int,B2: set_nat_set_int] :
( ! [A5: nat > set_int] :
( ( member_nat_set_int @ A5 @ A2 )
=> ? [X7: nat > set_int] :
( ( member_nat_set_int @ X7 @ B2 )
& ( ord_le3704955753469811889et_int @ A5 @ X7 ) ) )
=> ( ord_le3704955753469811889et_int @ ( comple93655007136625956et_int @ A2 ) @ ( comple93655007136625956et_int @ B2 ) ) ) ).
% Sup_mono
thf(fact_1193_Sup__mono,axiom,
! [A2: set_set_set_set_int,B2: set_set_set_set_int] :
( ! [A5: set_set_set_int] :
( ( member7356822600254261989et_int @ A5 @ A2 )
=> ? [X7: set_set_set_int] :
( ( member7356822600254261989et_int @ X7 @ B2 )
& ( ord_le4317611570275147438et_int @ A5 @ X7 ) ) )
=> ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ A2 ) @ ( comple1756060948637632417et_int @ B2 ) ) ) ).
% Sup_mono
thf(fact_1194_Sup__mono,axiom,
! [A2: set_set_set_int,B2: set_set_set_int] :
( ! [A5: set_set_int] :
( ( member_set_set_int @ A5 @ A2 )
=> ? [X7: set_set_int] :
( ( member_set_set_int @ X7 @ B2 )
& ( ord_le4403425263959731960et_int @ A5 @ X7 ) ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A2 ) @ ( comple7281953568134767595et_int @ B2 ) ) ) ).
% Sup_mono
thf(fact_1195_Sup__mono,axiom,
! [A2: set_set_int,B2: set_set_int] :
( ! [A5: set_int] :
( ( member_set_int @ A5 @ A2 )
=> ? [X7: set_int] :
( ( member_set_int @ X7 @ B2 )
& ( ord_less_eq_set_int @ A5 @ X7 ) ) )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A2 ) @ ( comple3221217463730067765et_int @ B2 ) ) ) ).
% Sup_mono
thf(fact_1196_Sup__least,axiom,
! [A2: set_nat_set_int,Z: nat > set_int] :
( ! [X4: nat > set_int] :
( ( member_nat_set_int @ X4 @ A2 )
=> ( ord_le3704955753469811889et_int @ X4 @ Z ) )
=> ( ord_le3704955753469811889et_int @ ( comple93655007136625956et_int @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_1197_Sup__least,axiom,
! [A2: set_set_set_set_int,Z: set_set_set_int] :
( ! [X4: set_set_set_int] :
( ( member7356822600254261989et_int @ X4 @ A2 )
=> ( ord_le4317611570275147438et_int @ X4 @ Z ) )
=> ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_1198_Sup__least,axiom,
! [A2: set_set_set_int,Z: set_set_int] :
( ! [X4: set_set_int] :
( ( member_set_set_int @ X4 @ A2 )
=> ( ord_le4403425263959731960et_int @ X4 @ Z ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_1199_Sup__least,axiom,
! [A2: set_set_int,Z: set_int] :
( ! [X4: set_int] :
( ( member_set_int @ X4 @ A2 )
=> ( ord_less_eq_set_int @ X4 @ Z ) )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_1200_Sup__upper,axiom,
! [X: nat > set_int,A2: set_nat_set_int] :
( ( member_nat_set_int @ X @ A2 )
=> ( ord_le3704955753469811889et_int @ X @ ( comple93655007136625956et_int @ A2 ) ) ) ).
% Sup_upper
thf(fact_1201_Sup__upper,axiom,
! [X: set_set_set_int,A2: set_set_set_set_int] :
( ( member7356822600254261989et_int @ X @ A2 )
=> ( ord_le4317611570275147438et_int @ X @ ( comple1756060948637632417et_int @ A2 ) ) ) ).
% Sup_upper
thf(fact_1202_Sup__upper,axiom,
! [X: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ X @ A2 )
=> ( ord_le4403425263959731960et_int @ X @ ( comple7281953568134767595et_int @ A2 ) ) ) ).
% Sup_upper
thf(fact_1203_Sup__upper,axiom,
! [X: set_int,A2: set_set_int] :
( ( member_set_int @ X @ A2 )
=> ( ord_less_eq_set_int @ X @ ( comple3221217463730067765et_int @ A2 ) ) ) ).
% Sup_upper
thf(fact_1204_Sup__le__iff,axiom,
! [A2: set_set_set_set_int,B: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ A2 ) @ B )
= ( ! [X2: set_set_set_int] :
( ( member7356822600254261989et_int @ X2 @ A2 )
=> ( ord_le4317611570275147438et_int @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1205_Sup__le__iff,axiom,
! [A2: set_set_set_int,B: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A2 ) @ B )
= ( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A2 )
=> ( ord_le4403425263959731960et_int @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1206_Sup__le__iff,axiom,
! [A2: set_set_int,B: set_int] :
( ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A2 ) @ B )
= ( ! [X2: set_int] :
( ( member_set_int @ X2 @ A2 )
=> ( ord_less_eq_set_int @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_1207_Sup__upper2,axiom,
! [U: nat > set_int,A2: set_nat_set_int,V: nat > set_int] :
( ( member_nat_set_int @ U @ A2 )
=> ( ( ord_le3704955753469811889et_int @ V @ U )
=> ( ord_le3704955753469811889et_int @ V @ ( comple93655007136625956et_int @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1208_Sup__upper2,axiom,
! [U: set_set_set_int,A2: set_set_set_set_int,V: set_set_set_int] :
( ( member7356822600254261989et_int @ U @ A2 )
=> ( ( ord_le4317611570275147438et_int @ V @ U )
=> ( ord_le4317611570275147438et_int @ V @ ( comple1756060948637632417et_int @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1209_Sup__upper2,axiom,
! [U: set_set_int,A2: set_set_set_int,V: set_set_int] :
( ( member_set_set_int @ U @ A2 )
=> ( ( ord_le4403425263959731960et_int @ V @ U )
=> ( ord_le4403425263959731960et_int @ V @ ( comple7281953568134767595et_int @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1210_Sup__upper2,axiom,
! [U: set_int,A2: set_set_int,V: set_int] :
( ( member_set_int @ U @ A2 )
=> ( ( ord_less_eq_set_int @ V @ U )
=> ( ord_less_eq_set_int @ V @ ( comple3221217463730067765et_int @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_1211_cSup__eq__maximum,axiom,
! [Z: set_int,X5: set_set_int] :
( ( member_set_int @ Z @ X5 )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ X5 )
=> ( ord_less_eq_set_int @ X4 @ Z ) )
=> ( ( comple3221217463730067765et_int @ X5 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_1212_int_Oquot__ideal__imp__ring__ideal,axiom,
! [I2: set_int,J2: set_set_int] :
( ( ideal_6787631597145370931t_unit @ I2 @ ( 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 ) ) ) )
=> ( ( ideal_7262958097527394045t_unit @ J2 @ ( 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 ) ) ) @ I2 ) )
=> ( ideal_6787631597145370931t_unit @ ( comple3221217463730067765et_int @ J2 ) @ ( 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_ideal_imp_ring_ideal
thf(fact_1213_int_Ocanonical__proj__vimage__in__carrier,axiom,
! [I2: set_int,J2: set_set_int] :
( ( ideal_6787631597145370931t_unit @ I2 @ ( 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 ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ J2 @ ( partia966996272515721803t_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 ) ) ) @ I2 ) ) )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ J2 ) @ top_top_set_int ) ) ) ).
% int.canonical_proj_vimage_in_carrier
thf(fact_1214_r_Ocanonical__proj__vimage__mem__iff,axiom,
! [I2: set_set_int,J2: set_set_set_int,A: set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ord_le4317611570275147438et_int @ J2 @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( comple7281953568134767595et_int @ J2 ) )
= ( member_set_set_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ A ) @ J2 ) ) ) ) ) ).
% r.canonical_proj_vimage_mem_iff
thf(fact_1215_r_Oline__extension__in__carrier,axiom,
! [K4: set_set_int,A: set_int,E2: set_set_int] :
( ( ord_le4403425263959731960et_int @ K4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ E2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( embedd4283282269743769663t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K4 @ A @ E2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.line_extension_in_carrier
thf(fact_1216_r_Oa__rcos__zero,axiom,
! [I2: set_set_int,I: set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ I @ I2 )
=> ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ I )
= I2 ) ) ) ).
% r.a_rcos_zero
thf(fact_1217_r_Oa__r__coset__subset__G,axiom,
! [H: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H @ X ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.a_r_coset_subset_G
thf(fact_1218_r_Oa__setmult__rcos__assoc,axiom,
! [H: set_set_int,K4: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K4 @ X ) )
= ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H @ K4 ) @ X ) ) ) ) ) ).
% r.a_setmult_rcos_assoc
thf(fact_1219_r_Oa__rcos__assoc__lcos,axiom,
! [H: set_set_int,K4: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H @ X ) @ K4 )
= ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ K4 ) ) ) ) ) ) ).
% r.a_rcos_assoc_lcos
thf(fact_1220_r_Oa__coset__add__zero,axiom,
! [M5: set_set_int] :
( ( ord_le4403425263959731960et_int @ M5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M5 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= M5 ) ) ).
% r.a_coset_add_zero
thf(fact_1221_r_Oquotient__eq__iff__same__a__r__cos,axiom,
! [I2: set_set_int,A: set_int,B: set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B ) @ I2 )
= ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ A )
= ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ B ) ) ) ) ) ) ).
% r.quotient_eq_iff_same_a_r_cos
thf(fact_1222_r_Oa__rcosetsI,axiom,
! [H: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_set_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H @ X ) @ ( a_RCOS5559887075240879033t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H ) ) ) ) ).
% r.a_rcosetsI
thf(fact_1223_r_Ominus__closed,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.minus_closed
thf(fact_1224_r_Or__right__minus__eq,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ B )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( A = B ) ) ) ) ).
% r.r_right_minus_eq
thf(fact_1225_int_Oa__coset__add__zero,axiom,
! [M5: set_int] :
( ( ord_less_eq_set_int @ M5 @ 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 ) ) ) @ M5 @ zero_zero_int )
= M5 ) ) ).
% int.a_coset_add_zero
thf(fact_1226_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_1227_int_Oa__coset__add__assoc,axiom,
! [M5: set_int,G2: int,H2: int] :
( ( ord_less_eq_set_int @ M5 @ top_top_set_int )
=> ( ( member_int @ G2 @ 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 ) ) ) @ M5 @ G2 ) @ 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 ) ) ) @ M5 @ ( plus_plus_int @ G2 @ H2 ) ) ) ) ) ) ).
% int.a_coset_add_assoc
thf(fact_1228_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_1229_int_Oa__rcos__zero,axiom,
! [I2: set_int,I: int] :
( ( ideal_6787631597145370931t_unit @ I2 @ ( 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 ) ) ) )
=> ( ( member_int @ I @ I2 )
=> ( ( 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 ) ) ) @ I2 @ I )
= I2 ) ) ) ).
% int.a_rcos_zero
thf(fact_1230_int_Ocanonical__proj__vimage__mem__iff,axiom,
! [I2: set_int,J2: set_set_int,A: int] :
( ( ideal_6787631597145370931t_unit @ I2 @ ( 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 ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ J2 @ ( partia966996272515721803t_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 ) ) ) @ I2 ) ) )
=> ( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ A @ ( comple3221217463730067765et_int @ J2 ) )
= ( 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 ) ) ) @ I2 @ A ) @ J2 ) ) ) ) ) ).
% int.canonical_proj_vimage_mem_iff
thf(fact_1231_zfact__coset,axiom,
! [N: nat,X: set_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N ) ) ) )
=> ( 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 ) ) ) @ ( 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 ) ) @ ( semiri1314217659103216013at_int @ ( zfact_iso_inv @ N @ X ) ) ) ) ) ) ).
% zfact_coset
thf(fact_1232_r_Orcosets__subset__PowG,axiom,
! [H: set_set_int] :
( ( additi7073586575563672860t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ord_le4317611570275147438et_int @ ( a_RCOS5559887075240879033t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H ) @ ( pow_set_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.rcosets_subset_PowG
thf(fact_1233_zfact__iso__inv__0,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( zfact_iso_inv @ N @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N ) ) ) )
= zero_zero_nat ) ) ).
% zfact_iso_inv_0
thf(fact_1234_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_1235_r_Oring__ideal__imp__quot__ideal,axiom,
! [I2: set_set_int,J2: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ideal_7262958097527394045t_unit @ J2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ideal_1992573625310827463t_unit @ ( image_1010086626112315521et_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 ) @ J2 ) @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 ) ) ) ) ).
% r.ring_ideal_imp_quot_ideal
thf(fact_1236_rcos__zfact,axiom,
! [K: int,L: int,R: int] :
( ( member_int @ K @ ( zMod @ L @ R ) )
=> ? [X4: int] :
( K
= ( plus_plus_int @ ( times_times_int @ X4 @ L ) @ R ) ) ) ).
% rcos_zfact
thf(fact_1237_int_Oring__ideal__imp__quot__ideal,axiom,
! [I2: set_int,J2: set_int] :
( ( ideal_6787631597145370931t_unit @ I2 @ ( 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 ) ) ) )
=> ( ( ideal_6787631597145370931t_unit @ J2 @ ( 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 ) ) ) )
=> ( ideal_7262958097527394045t_unit @ ( image_int_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 ) ) ) @ I2 ) @ J2 ) @ ( 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 ) ) ) @ I2 ) ) ) ) ).
% int.ring_ideal_imp_quot_ideal
thf(fact_1238_r_Ogenideal__one,axiom,
( ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( insert_set_int @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
= ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.genideal_one
thf(fact_1239_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_1240_r_Ocarrier__one__not__zero,axiom,
( ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
= ( ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.carrier_one_not_zero
thf(fact_1241_r_Ocarrier__one__zero,axiom,
( ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
= ( ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.carrier_one_zero
thf(fact_1242_r_Oone__zeroD,axiom,
( ( ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) ) ) ).
% r.one_zeroD
thf(fact_1243_r_Oone__zeroI,axiom,
( ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
=> ( ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.one_zeroI
thf(fact_1244_r_Oone__closed,axiom,
member_set_int @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% r.one_closed
thf(fact_1245_zfact__iso__0,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ring_zfact_iso @ N @ zero_zero_nat )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% zfact_iso_0
thf(fact_1246_r_Ocring__fieldI2,axiom,
( ( ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ! [A5: set_int] :
( ( member_set_int @ A5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( A5
!= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [X7: set_int] :
( ( member_set_int @ X7 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A5 @ X7 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) )
=> ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.cring_fieldI2
thf(fact_1247_r_Om__lcomm,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Z ) ) ) ) ) ) ).
% r.m_lcomm
thf(fact_1248_r_Om__comm,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X ) ) ) ) ).
% r.m_comm
thf(fact_1249_r_Om__assoc,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) ) ) ) ) ) ).
% r.m_assoc
thf(fact_1250_r_Oinv__unique,axiom,
! [Y: set_int,X: set_int,Y2: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y2 )
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% r.inv_unique
thf(fact_1251_r_Oone__unique,axiom,
! [U: set_int] :
( ( member_set_int @ U @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ U @ X4 )
= X4 ) )
=> ( U
= ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.one_unique
thf(fact_1252_r_Om__closed,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.m_closed
thf(fact_1253_r_Ol__null,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.l_null
thf(fact_1254_r_Or__null,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.r_null
thf(fact_1255_r_Or__one,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= X ) ) ).
% r.r_one
thf(fact_1256_r_Ol__one,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( one_se8065767436706823081t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X )
= X ) ) ).
% r.l_one
thf(fact_1257_int_Oadd__additive__subgroups,axiom,
! [H: set_int,K4: set_int] :
( ( additi1413919225584036690t_unit @ H @ ( 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 ) ) ) )
=> ( ( additi1413919225584036690t_unit @ K4 @ ( 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 ) ) ) )
=> ( additi1413919225584036690t_unit @ ( set_ad6660713454305125854t_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 @ K4 ) @ ( 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.add_additive_subgroups
thf(fact_1258_int_Oset__add__comm,axiom,
! [I2: set_int,J2: set_int] :
( ( ord_less_eq_set_int @ I2 @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ J2 @ top_top_set_int )
=> ( ( set_ad6660713454305125854t_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 ) ) ) @ I2 @ J2 )
= ( set_ad6660713454305125854t_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 ) ) ) @ J2 @ I2 ) ) ) ) ).
% int.set_add_comm
thf(fact_1259_int_Oset__add__closed,axiom,
! [A2: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A2 @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ B2 @ top_top_set_int )
=> ( ord_less_eq_set_int @ ( set_ad6660713454305125854t_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 ) ) ) @ A2 @ B2 ) @ top_top_set_int ) ) ) ).
% int.set_add_closed
thf(fact_1260_int_Osetadd__subset__G,axiom,
! [H: set_int,K4: set_int] :
( ( ord_less_eq_set_int @ H @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ K4 @ top_top_set_int )
=> ( ord_less_eq_set_int @ ( set_ad6660713454305125854t_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 @ K4 ) @ top_top_set_int ) ) ) ).
% int.setadd_subset_G
thf(fact_1261_int_Oadd__ideals,axiom,
! [I2: set_int,J2: set_int] :
( ( ideal_6787631597145370931t_unit @ I2 @ ( 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 ) ) ) )
=> ( ( ideal_6787631597145370931t_unit @ J2 @ ( 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 ) ) ) )
=> ( ideal_6787631597145370931t_unit @ ( set_ad6660713454305125854t_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 ) ) ) @ I2 @ J2 ) @ ( 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.add_ideals
thf(fact_1262_int_Oa__setmult__rcos__assoc,axiom,
! [H: set_int,K4: set_int,X: int] :
( ( ord_less_eq_set_int @ H @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ K4 @ top_top_set_int )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( set_ad6660713454305125854t_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 @ ( 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 ) ) ) @ K4 @ 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 ) ) ) @ ( set_ad6660713454305125854t_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 @ K4 ) @ X ) ) ) ) ) ).
% int.a_setmult_rcos_assoc
thf(fact_1263_int_Orcosets__subset__PowG,axiom,
! [H: set_int] :
( ( additi1413919225584036690t_unit @ H @ ( 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 ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( 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 ) @ ( pow_int @ top_top_set_int ) ) ) ).
% int.rcosets_subset_PowG
thf(fact_1264_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_1265_r_Omonoid__cancelI,axiom,
( ! [A5: set_int,B6: set_int,C5: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C5 @ A5 )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C5 @ B6 ) )
=> ( ( member_set_int @ A5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B6 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( A5 = B6 ) ) ) ) )
=> ( ! [A5: set_int,B6: set_int,C5: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A5 @ C5 )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B6 @ C5 ) )
=> ( ( member_set_int @ A5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B6 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( A5 = B6 ) ) ) ) )
=> ( monoid497721730651901107t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.monoid_cancelI
thf(fact_1266_r_Oa__lcos__m__assoc,axiom,
! [M5: set_set_int,G2: set_int,H2: set_int] :
( ( ord_le4403425263959731960et_int @ M5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ G2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ G2 @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 @ M5 ) )
= ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ G2 @ H2 ) @ M5 ) ) ) ) ) ).
% r.a_lcos_m_assoc
thf(fact_1267_r_Oadd_Ol__cancel,axiom,
! [C: set_int,A: set_int,B: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C @ A )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C @ B ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( A = B ) ) ) ) ) ).
% r.add.l_cancel
thf(fact_1268_r_Oadd_Om__assoc,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) ) ) ) ) ) ).
% r.add.m_assoc
thf(fact_1269_r_Oadd_Om__comm,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X ) ) ) ) ).
% r.add.m_comm
% Conjectures (1)
thf(conj_0,conjecture,
ideal_6787631597145370931t_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 @ ( semiri1314217659103216013at_int @ n ) @ 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 ) ) ) ).
%------------------------------------------------------------------------------