TPTP Problem File: SLH0487^1.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_00141_004986__18280732_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1568 ( 450 unt; 291 typ; 0 def)
% Number of atoms : 3809 (1435 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 17061 ( 234 ~; 37 |; 139 &;14866 @)
% ( 0 <=>;1785 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Number of types : 29 ( 28 usr)
% Number of type conns : 1070 (1070 >; 0 *; 0 +; 0 <<)
% Number of symbols : 266 ( 263 usr; 26 con; 0-4 aty)
% Number of variables : 3102 ( 22 ^;2970 !; 110 ?;3102 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:41:22.021
%------------------------------------------------------------------------------
% Could-be-implicit typings (28)
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% Explicit typings (263)
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member_set_int_int: ( set_int > int ) > set_set_int_int > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
member5205197933313416826et_int: ( set_int > set_int ) > set_set_int_set_int2 > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
member1123395007370249776et_int: ( set_int > set_set_int ) > set_se3762516360063217359et_int > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Int__Oint_J,type,
member2922251196090340986nt_int: ( set_set_int > int ) > set_set_set_int_int > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
member2198268278526348592et_int: ( set_set_int > set_int ) > set_se7044434731094029775et_int > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
member5110176937812460956et_int: ( set_set_int > set_set_set_int ) > set_se2452231264655218747et_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_I_062_It__Int__Oint_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
member7905727001773941114et_int: set_int_set_int > set_set_int_set_int > $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 ).
thf(sy_v_x_H____,type,
x: nat ).
thf(sy_v_x____,type,
x2: set_int ).
thf(sy_v_y_H____,type,
y: nat ).
thf(sy_v_y____,type,
y2: set_int ).
% Relevant facts (1273)
thf(fact_0_i_OIcarr,axiom,
! [I: int] :
( ( member_int @ I @ i )
=> ( member_int @ I @ top_top_set_int ) ) ).
% i.Icarr
thf(fact_1_i_Oone__imp__carrier,axiom,
( ( member_int @ one_one_int @ i )
=> ( i = top_top_set_int ) ) ).
% i.one_imp_carrier
thf(fact_2_i_OI__l__closed,axiom,
! [A: int,X: int] :
( ( member_int @ A @ i )
=> ( ( member_int @ X @ top_top_set_int )
=> ( member_int @ ( times_times_int @ X @ A ) @ i ) ) ) ).
% i.I_l_closed
thf(fact_3_i_OI__r__closed,axiom,
! [A: int,X: int] :
( ( member_int @ A @ i )
=> ( ( member_int @ X @ top_top_set_int )
=> ( member_int @ ( times_times_int @ A @ X ) @ i ) ) ) ).
% i.I_r_closed
thf(fact_4_i_Ohelper__I__closed,axiom,
! [A: int,X: int,Y: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ ( times_times_int @ A @ X ) @ i )
=> ( member_int @ ( times_times_int @ A @ ( times_times_int @ X @ Y ) ) @ i ) ) ) ) ) ).
% i.helper_I_closed
thf(fact_5_x_H__def,axiom,
( x
= ( zfact_iso_inv @ n @ x2 ) ) ).
% x'_def
thf(fact_6_y_H__def,axiom,
( y
= ( zfact_iso_inv @ n @ y2 ) ) ).
% y'_def
thf(fact_7_i_Oa__elemrcos__carrier,axiom,
! [A: int,A2: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ A2 @ ( 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 ) ) ) @ i @ A ) )
=> ( member_int @ A2 @ top_top_set_int ) ) ) ).
% i.a_elemrcos_carrier
thf(fact_8_i_Oa__rcos__const,axiom,
! [H: int] :
( ( member_int @ H @ i )
=> ( ( 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 ) ) ) @ i @ H )
= i ) ) ).
% i.a_rcos_const
thf(fact_9_i_Oa__rcos__self,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( member_int @ 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 ) ) ) @ i @ X ) ) ) ).
% i.a_rcos_self
thf(fact_10_i_Oa__repr__independence_H,axiom,
! [Y: int,X: int] :
( ( member_int @ Y @ ( 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 ) ) ) @ i @ X ) )
=> ( ( member_int @ X @ 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 ) ) ) @ i @ 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 ) ) ) @ i @ Y ) ) ) ) ).
% i.a_repr_independence'
thf(fact_11_i_Oa__repr__independenceD,axiom,
! [Y: int,X: int] :
( ( member_int @ Y @ 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 ) ) ) @ i @ 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 ) ) ) @ i @ Y ) )
=> ( member_int @ Y @ ( 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 ) ) ) @ i @ X ) ) ) ) ).
% i.a_repr_independenceD
thf(fact_12_i_Orcos__const__imp__mem,axiom,
! [I: int] :
( ( member_int @ I @ 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 ) ) ) @ i @ I )
= i )
=> ( member_int @ I @ i ) ) ) ).
% i.rcos_const_imp_mem
thf(fact_13_x__carr,axiom,
member_set_int @ x2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% x_carr
thf(fact_14_y__carr,axiom,
member_set_int @ y2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% y_carr
thf(fact_15_i_Orcoset__mult__add,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( rcoset2619718275356395046t_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 @ ( 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 ) ) ) @ i @ 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 ) ) ) @ i @ Y ) )
= ( 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 ) ) ) @ i @ ( times_times_int @ X @ Y ) ) ) ) ) ).
% i.rcoset_mult_add
thf(fact_16_s_Ohom__add,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ 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 ) ) ) @ i @ ( plus_plus_int @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i @ 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 ) ) ) @ i @ Y ) ) ) ) ) ).
% s.hom_add
thf(fact_17_s_Ois__abelian__group__hom,axiom,
abelia4252751761703478237t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) ).
% s.is_abelian_group_hom
thf(fact_18_s_Ohomh,axiom,
member_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 ) ) ) @ i ) @ ( ring_h1577560673974712708t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% s.homh
thf(fact_19_s_Ohom__closed,axiom,
! [X: 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 ) ) ) @ i @ X ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.hom_closed
thf(fact_20_s_Ozero__closed,axiom,
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 ) ) ) @ i @ zero_zero_int ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% s.zero_closed
thf(fact_21_i_Ois__additive__subgroup,axiom,
additi1413919225584036690t_unit @ 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 ) ) ) ).
% i.is_additive_subgroup
thf(fact_22_i_Oa__subgroup__in__rcosets,axiom,
member_set_int @ i @ ( 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 ) ) ) @ i ) ).
% i.a_subgroup_in_rcosets
thf(fact_23_i_Ois__abelian__subgroup,axiom,
abelia4388627187832904468t_unit @ 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 ) ) ) ).
% i.is_abelian_subgroup
thf(fact_24_i_Oa__coset__eq,axiom,
! [X2: int] :
( ( member_int @ X2 @ 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 ) ) ) @ i @ X2 )
= ( a_l_co457318074463736598t_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 ) ) ) @ X2 @ i ) ) ) ).
% i.a_coset_eq
thf(fact_25_s_Ohomeq__imp__rcos,axiom,
! [A: int,X: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ X @ 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 ) ) ) @ i @ 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 ) ) ) @ i @ A ) )
=> ( member_int @ 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 ) ) ) @ ( a_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) ) @ A ) ) ) ) ) ).
% s.homeq_imp_rcos
thf(fact_26_s_Orcos__imp__homeq,axiom,
! [A: int,X: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ 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 ) ) ) @ ( a_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) ) @ A ) )
=> ( ( 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 ) ) ) @ i @ 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 ) ) ) @ i @ A ) ) ) ) ).
% s.rcos_imp_homeq
thf(fact_27_s_Oring__hom__ring__axioms,axiom,
ring_h4095663460228855494t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) ).
% s.ring_hom_ring_axioms
thf(fact_28_i_Oa__Hcarr,axiom,
! [H: int] :
( ( member_int @ H @ i )
=> ( member_int @ H @ ( 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 ) ) ) ) ) ) ).
% i.a_Hcarr
thf(fact_29_r_Oadd_Or__cancel,axiom,
! [A: set_int,C: set_int,B: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ C ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( A = B ) ) ) ) ) ).
% r.add.r_cancel
thf(fact_30_r_Oadd_Om__lcomm,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Z ) ) ) ) ) ) ).
% r.add.m_lcomm
thf(fact_31_r_Oadd_Om__comm,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X ) ) ) ) ).
% r.add.m_comm
thf(fact_32_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_33_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_34_s_Oadditive__subgroup__a__kernel,axiom,
additi1413919225584036690t_unit @ ( a_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ 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 ) ) ) ).
% s.additive_subgroup_a_kernel
thf(fact_35_s_Oabelian__subgroup__a__kernel,axiom,
abelia4388627187832904468t_unit @ ( a_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ 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 ) ) ) ).
% s.abelian_subgroup_a_kernel
thf(fact_36_r_Oadd_Oright__cancel,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% r.add.right_cancel
thf(fact_37_r_Oadd_Om__closed,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.m_closed
thf(fact_38_i_Oa__setinv__closed,axiom,
! [K: set_int] :
( ( member_set_int @ K @ ( 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 ) ) ) @ i ) )
=> ( member_set_int @ ( a_SET_5361721377625069265t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K ) @ ( 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 ) ) ) @ i ) ) ) ).
% i.a_setinv_closed
thf(fact_39_int__Zcarr,axiom,
! [K2: int] : ( member_int @ K2 @ ( 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_40_r_Oonepideal,axiom,
princi8860937869964495385t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.onepideal
thf(fact_41_s_Oquot__mem,axiom,
! [X3: set_int] :
( ( member_set_int @ X3 @ ( 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 ) ) ) @ ( a_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) ) ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ top_top_set_int )
& ( X3
= ( 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_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) ) @ X4 ) ) ) ) ).
% s.quot_mem
thf(fact_42_s_Oa__inv__closed,axiom,
! [X: 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 ) ) ) @ i @ ( uminus_uminus_int @ X ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.a_inv_closed
thf(fact_43_int_Oring__hom__restrict,axiom,
! [F: int > set_int,S: partia4934656038542163276t_unit,G: int > set_int] :
( ( member_int_set_int @ F @ ( ring_h1577560673974712708t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ S ) )
=> ( ! [R: int] :
( ( member_int @ R @ top_top_set_int )
=> ( ( F @ R )
= ( G @ R ) ) )
=> ( member_int_set_int @ G @ ( ring_h1577560673974712708t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ S ) ) ) ) ).
% int.ring_hom_restrict
thf(fact_44_i_Orcos__ring__hom,axiom,
member_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 ) ) ) @ i ) @ ( ring_h1577560673974712708t_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 ) ) ) @ i ) ) ).
% i.rcos_ring_hom
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: int > set_int,P: ( int > set_int ) > $o] :
( ( member_int_set_int @ A @ ( collect_int_set_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_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_49_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A3: set_int] :
( ( collect_int
@ ^ [X5: int] : ( member_int @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A3: set_set_int] :
( ( collect_set_int
@ ^ [X5: set_int] : ( member_set_int @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A3: set_int_set_int] :
( ( collect_int_set_int
@ ^ [X5: int > set_int] : ( member_int_set_int @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
! [A3: set_set_set_int] :
( ( collect_set_set_int
@ ^ [X5: set_set_int] : ( member_set_set_int @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X5: nat] : ( member_nat @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_55_r_Oint__embed__ring__hom,axiom,
ring_h4095663460228855494t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% r.int_embed_ring_hom
thf(fact_56_i_Orcos__ring__hom__ring,axiom,
ring_h4095663460228855494t_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 ) ) ) @ i ) @ ( 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 ) ) ) @ i ) ).
% i.rcos_ring_hom_ring
thf(fact_57_s_Okernel__is__ideal,axiom,
ideal_6787631597145370931t_unit @ ( a_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ 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 ) ) ) ).
% s.kernel_is_ideal
thf(fact_58_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_59_i_Oa__rcosets__carrier,axiom,
! [X3: set_int] :
( ( member_set_int @ X3 @ ( 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 ) ) ) @ i ) )
=> ( ord_less_eq_set_int @ X3 @ top_top_set_int ) ) ).
% i.a_rcosets_carrier
thf(fact_60_UNIV_I4_J,axiom,
! [P: int > $o] :
( ( ? [X5: int] :
( ( member_int @ X5 @ top_top_set_int )
& ( P @ X5 ) ) )
= ( ? [X6: int] : ( P @ X6 ) ) ) ).
% UNIV(4)
thf(fact_61_UNIV_I4_J,axiom,
! [P: nat > $o] :
( ( ? [X5: nat] :
( ( member_nat @ X5 @ top_top_set_nat )
& ( P @ X5 ) ) )
= ( ? [X6: nat] : ( P @ X6 ) ) ) ).
% UNIV(4)
thf(fact_62_UNIV_I3_J,axiom,
! [P: int > $o] :
( ( ! [X5: int] :
( ( member_int @ X5 @ top_top_set_int )
=> ( P @ X5 ) ) )
= ( ! [X6: int] : ( P @ X6 ) ) ) ).
% UNIV(3)
thf(fact_63_UNIV_I3_J,axiom,
! [P: nat > $o] :
( ( ! [X5: nat] :
( ( member_nat @ X5 @ top_top_set_nat )
=> ( P @ X5 ) ) )
= ( ! [X6: nat] : ( P @ X6 ) ) ) ).
% UNIV(3)
thf(fact_64_i_Oa__inv__op__closed1,axiom,
! [X: int,H: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ H @ i )
=> ( member_int @ ( plus_plus_int @ ( plus_plus_int @ ( uminus_uminus_int @ X ) @ H ) @ X ) @ i ) ) ) ).
% i.a_inv_op_closed1
thf(fact_65_i_Oa__inv__op__closed2,axiom,
! [X: int,H: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ H @ i )
=> ( member_int @ ( plus_plus_int @ ( plus_plus_int @ X @ H ) @ ( uminus_uminus_int @ X ) ) @ i ) ) ) ).
% i.a_inv_op_closed2
thf(fact_66_r_Oint__embed__closed,axiom,
! [K2: int] : ( member_set_int @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.int_embed_closed
thf(fact_67_r_Oint__embed__add,axiom,
! [X: int,Y: int] :
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( plus_plus_int @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ).
% r.int_embed_add
thf(fact_68_int_Oadd_Oone__closed,axiom,
member_int @ zero_zero_int @ top_top_set_int ).
% int.add.one_closed
thf(fact_69_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_70_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_71_int_Oone__closed,axiom,
member_int @ one_one_int @ top_top_set_int ).
% int.one_closed
thf(fact_72_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_73_int_Ominus__minus,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( uminus_uminus_int @ ( uminus_uminus_int @ X ) )
= X ) ) ).
% int.minus_minus
thf(fact_74_int_Oadd_Oinv__closed,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( member_int @ ( uminus_uminus_int @ X ) @ top_top_set_int ) ) ).
% int.add.inv_closed
thf(fact_75_int_Ominus__zero,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% int.minus_zero
thf(fact_76_i_Ois__ideal,axiom,
ideal_6787631597145370931t_unit @ 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 ) ) ) ).
% i.is_ideal
thf(fact_77_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_78_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_79_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_80_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_81_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_82_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_83_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_84_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_85_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_86_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_87_int_Oadd_Oinv__eq__1__iff,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( ( uminus_uminus_int @ X )
= zero_zero_int )
= ( X = zero_zero_int ) ) ) ).
% int.add.inv_eq_1_iff
thf(fact_88_i_Oa__subset,axiom,
ord_less_eq_set_int @ i @ ( 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 ) ) ) ) ).
% i.a_subset
thf(fact_89_i_Oa__rcos__module,axiom,
! [X: int,X7: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ X7 @ top_top_set_int )
=> ( ( member_int @ X7 @ ( 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 ) ) ) @ i @ X ) )
= ( member_int @ ( plus_plus_int @ X7 @ ( uminus_uminus_int @ X ) ) @ i ) ) ) ) ).
% i.a_rcos_module
thf(fact_90_i_Oa__rcos__module__imp,axiom,
! [X: int,X7: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ X7 @ ( 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 ) ) ) @ i @ X ) )
=> ( member_int @ ( plus_plus_int @ X7 @ ( uminus_uminus_int @ X ) ) @ i ) ) ) ).
% i.a_rcos_module_imp
thf(fact_91_i_Oa__rcos__module__rev,axiom,
! [X: int,X7: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ X7 @ top_top_set_int )
=> ( ( member_int @ ( plus_plus_int @ X7 @ ( uminus_uminus_int @ X ) ) @ i )
=> ( member_int @ X7 @ ( 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 ) ) ) @ i @ X ) ) ) ) ) ).
% i.a_rcos_module_rev
thf(fact_92_i_Oa__rcos__inv,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( a_SET_5361721377625069265t_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 ) ) ) @ i @ 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 ) ) ) @ i @ ( uminus_uminus_int @ X ) ) ) ) ).
% i.a_rcos_inv
thf(fact_93_int_Oa__coset__add__zero,axiom,
! [M: set_int] :
( ( ord_less_eq_set_int @ M @ top_top_set_int )
=> ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ zero_zero_int )
= M ) ) ).
% int.a_coset_add_zero
thf(fact_94_UNIV_I2_J,axiom,
! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ top_top_set_nat ) ).
% UNIV(2)
thf(fact_95_UNIV_I2_J,axiom,
! [A3: set_int] : ( ord_less_eq_set_int @ A3 @ top_top_set_int ) ).
% UNIV(2)
thf(fact_96_UNIV_I2_J,axiom,
! [A3: set_set_int] : ( ord_le4403425263959731960et_int @ A3 @ top_top_set_set_int ) ).
% UNIV(2)
thf(fact_97_UNIV_I2_J,axiom,
! [A3: set_set_set_int] : ( ord_le4317611570275147438et_int @ A3 @ top_to5524576366173240574et_int ) ).
% UNIV(2)
thf(fact_98_int_Or__neg1,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( plus_plus_int @ ( uminus_uminus_int @ X ) @ ( plus_plus_int @ X @ Y ) )
= Y ) ) ) ).
% int.r_neg1
thf(fact_99_int_Or__neg2,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( plus_plus_int @ X @ ( plus_plus_int @ ( uminus_uminus_int @ X ) @ Y ) )
= Y ) ) ) ).
% int.r_neg2
thf(fact_100_int_Oadd_Oinv__mult,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( uminus_uminus_int @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) ) ).
% int.add.inv_mult
thf(fact_101_int_Oa__transpose__inv,axiom,
! [X: int,Y: int,Z: int] :
( ( ( plus_plus_int @ X @ Y )
= Z )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( member_int @ Z @ top_top_set_int )
=> ( ( plus_plus_int @ ( uminus_uminus_int @ X ) @ Z )
= Y ) ) ) ) ) ).
% int.a_transpose_inv
thf(fact_102_int_Oadd_Oinv__mult__group,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( uminus_uminus_int @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( uminus_uminus_int @ Y ) @ ( uminus_uminus_int @ X ) ) ) ) ) ).
% int.add.inv_mult_group
thf(fact_103_int_Oadd_Oinv__solve__left,axiom,
! [A: int,B: int,C: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C @ top_top_set_int )
=> ( ( A
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ C ) )
= ( C
= ( plus_plus_int @ B @ A ) ) ) ) ) ) ).
% int.add.inv_solve_left
thf(fact_104_int_Oadd_Oinv__solve__left_H,axiom,
! [A: int,B: int,C: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C @ top_top_set_int )
=> ( ( ( plus_plus_int @ ( uminus_uminus_int @ B ) @ C )
= A )
= ( C
= ( plus_plus_int @ B @ A ) ) ) ) ) ) ).
% int.add.inv_solve_left'
thf(fact_105_int_Oadd_Oinv__solve__right,axiom,
! [A: int,B: int,C: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C @ top_top_set_int )
=> ( ( A
= ( plus_plus_int @ B @ ( uminus_uminus_int @ C ) ) )
= ( B
= ( plus_plus_int @ A @ C ) ) ) ) ) ) ).
% int.add.inv_solve_right
thf(fact_106_int_Oadd_Oinv__solve__right_H,axiom,
! [A: int,B: int,C: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ C @ top_top_set_int )
=> ( ( ( plus_plus_int @ B @ ( uminus_uminus_int @ C ) )
= A )
= ( B
= ( plus_plus_int @ A @ C ) ) ) ) ) ) ).
% int.add.inv_solve_right'
thf(fact_107_int_Ol__minus,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( times_times_int @ ( uminus_uminus_int @ X ) @ Y )
= ( uminus_uminus_int @ ( times_times_int @ X @ Y ) ) ) ) ) ).
% int.l_minus
thf(fact_108_int_Or__minus,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( times_times_int @ X @ ( uminus_uminus_int @ Y ) )
= ( uminus_uminus_int @ ( times_times_int @ X @ Y ) ) ) ) ) ).
% int.r_minus
thf(fact_109_int_Oadd_Ol__inv,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( plus_plus_int @ ( uminus_uminus_int @ X ) @ X )
= zero_zero_int ) ) ).
% int.add.l_inv
thf(fact_110_int_Oadd_Or__inv,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( plus_plus_int @ X @ ( uminus_uminus_int @ X ) )
= zero_zero_int ) ) ).
% int.add.r_inv
thf(fact_111_int_Osum__zero__eq__neg,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
=> ( X
= ( uminus_uminus_int @ Y ) ) ) ) ) ).
% int.sum_zero_eq_neg
thf(fact_112_int_Oadd_Oinv__equality,axiom,
! [Y: int,X: int] :
( ( ( plus_plus_int @ Y @ X )
= zero_zero_int )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( uminus_uminus_int @ X )
= Y ) ) ) ) ).
% int.add.inv_equality
thf(fact_113_int_Osquare__eq__one,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( ( times_times_int @ X @ X )
= one_one_int )
=> ( ( X = one_one_int )
| ( X
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% int.square_eq_one
thf(fact_114_int_Oa__coset__add__inv1,axiom,
! [M: set_int,X: int,Y: int] :
( ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ ( plus_plus_int @ X @ ( uminus_uminus_int @ Y ) ) )
= M )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ M @ top_top_set_int )
=> ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ 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 ) ) ) @ M @ Y ) ) ) ) ) ) ).
% int.a_coset_add_inv1
thf(fact_115_int_Oa__coset__add__inv2,axiom,
! [M: set_int,X: int,Y: int] :
( ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ 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 ) ) ) @ M @ Y ) )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ M @ top_top_set_int )
=> ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ ( plus_plus_int @ X @ ( uminus_uminus_int @ Y ) ) )
= M ) ) ) ) ) ).
% int.a_coset_add_inv2
thf(fact_116_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_117_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_118_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_119_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_120_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_121_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_122_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_123_int_Ozero__not__one,axiom,
zero_zero_int != one_one_int ).
% int.zero_not_one
thf(fact_124_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_125_int_Oa__rcosI,axiom,
! [H: int,H2: set_int,X: int] :
( ( member_int @ H @ H2 )
=> ( ( ord_less_eq_set_int @ H2 @ top_top_set_int )
=> ( ( member_int @ X @ top_top_set_int )
=> ( member_int @ ( plus_plus_int @ H @ 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 ) ) ) @ H2 @ X ) ) ) ) ) ).
% int.a_rcosI
thf(fact_126_int_Oa__coset__add__assoc,axiom,
! [M: set_int,G: int,H: int] :
( ( ord_less_eq_set_int @ M @ top_top_set_int )
=> ( ( member_int @ G @ top_top_set_int )
=> ( ( member_int @ H @ top_top_set_int )
=> ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ G ) @ 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 ) ) ) @ M @ ( plus_plus_int @ G @ H ) ) ) ) ) ) ).
% int.a_coset_add_assoc
thf(fact_127_int_Oa__r__coset__subset__G,axiom,
! [H2: set_int,X: int] :
( ( ord_less_eq_set_int @ H2 @ 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 ) ) ) @ H2 @ X ) @ top_top_set_int ) ) ) ).
% int.a_r_coset_subset_G
thf(fact_128_int_Oa__lcos__m__assoc,axiom,
! [M: set_int,G: int,H: int] :
( ( ord_less_eq_set_int @ M @ top_top_set_int )
=> ( ( member_int @ G @ top_top_set_int )
=> ( ( member_int @ H @ top_top_set_int )
=> ( ( a_l_co457318074463736598t_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 ) ) ) @ G @ ( a_l_co457318074463736598t_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 @ M ) )
= ( a_l_co457318074463736598t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( plus_plus_int @ G @ H ) @ M ) ) ) ) ) ).
% int.a_lcos_m_assoc
thf(fact_129_int_Oa__lcos__mult__one,axiom,
! [M: set_int] :
( ( ord_less_eq_set_int @ M @ top_top_set_int )
=> ( ( a_l_co457318074463736598t_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 @ M )
= M ) ) ).
% int.a_lcos_mult_one
thf(fact_130_int_Oa__l__coset__subset__G,axiom,
! [H2: set_int,X: int] :
( ( ord_less_eq_set_int @ H2 @ top_top_set_int )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ord_less_eq_set_int @ ( a_l_co457318074463736598t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ H2 ) @ top_top_set_int ) ) ) ).
% int.a_l_coset_subset_G
thf(fact_131_int_Oa__rcosetsI,axiom,
! [H2: set_int,X: int] :
( ( ord_less_eq_set_int @ H2 @ 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 ) ) ) @ H2 @ 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 ) ) ) @ H2 ) ) ) ) ).
% int.a_rcosetsI
thf(fact_132_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_133_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_134_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_135_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_136_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_137_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_138_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_139_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_140_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_141_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_142_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_143_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_144_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_145_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_146_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_147_i_Oa__cosets__finite,axiom,
! [C: set_int] :
( ( member_set_int @ C @ ( 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 ) ) ) @ i ) )
=> ( ( ord_less_eq_set_int @ i @ top_top_set_int )
=> ( ( finite_finite_int @ top_top_set_int )
=> ( finite_finite_int @ C ) ) ) ) ).
% i.a_cosets_finite
thf(fact_148_i_Oadd_Ocarrier__not__empty,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 ) ) ) )
!= bot_bot_set_int ) ).
% i.add.carrier_not_empty
thf(fact_149_i_Oa__rcosets__inv__mult__group__eq,axiom,
! [M: set_int] :
( ( member_set_int @ M @ ( 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 ) ) ) @ i ) )
=> ( ( 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 ) ) ) @ ( a_SET_5361721377625069265t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M ) @ M )
= i ) ) ).
% i.a_rcosets_inv_mult_group_eq
thf(fact_150_i_Oa__rcosets__part__G,axiom,
( ( comple3221217463730067765et_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 ) ) ) @ i ) )
= top_top_set_int ) ).
% i.a_rcosets_part_G
thf(fact_151_i_Ozero__closed,axiom,
member_int @ ( 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 ) ) ) ) @ i ).
% i.zero_closed
thf(fact_152_i_Oquotient__is__ring,axiom,
ring_s5316885176909347197t_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 ) ) ) @ i ) ).
% i.quotient_is_ring
thf(fact_153_i_Oadd_Ol__cancel,axiom,
! [C: int,A: int,B: int] :
( ( ( add_int_Product_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 ) ) ) @ C @ A )
= ( add_int_Product_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 ) ) ) @ C @ B ) )
=> ( ( member_int @ A @ ( 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 ) ) ) ) )
=> ( ( member_int @ B @ ( 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 ) ) ) ) )
=> ( ( member_int @ C @ ( 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 ) ) ) ) )
=> ( A = B ) ) ) ) ) ).
% i.add.l_cancel
thf(fact_154_i_Oadd_Om__assoc,axiom,
! [X: int,Y: int,Z: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( member_int @ Y @ ( 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 ) ) ) ) )
=> ( ( member_int @ Z @ ( 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 ) ) ) ) )
=> ( ( add_int_Product_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 ) ) ) @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ Y ) @ Z )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ Z ) ) ) ) ) ) ).
% i.add.m_assoc
thf(fact_155_i_Oadd_Om__comm,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( member_int @ Y @ ( 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 ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ Y )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ X ) ) ) ) ).
% i.add.m_comm
thf(fact_156_r_Oring__axioms,axiom,
ring_s5316885176909347197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.ring_axioms
thf(fact_157_n__ge__1,axiom,
ord_less_nat @ one_one_nat @ n ).
% n_ge_1
thf(fact_158_i_Oadd_Or__cancel,axiom,
! [A: int,C: int,B: int] :
( ( ( add_int_Product_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 @ C )
= ( add_int_Product_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 ) ) ) @ B @ C ) )
=> ( ( member_int @ A @ ( 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 ) ) ) ) )
=> ( ( member_int @ B @ ( 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 ) ) ) ) )
=> ( ( member_int @ C @ ( 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 ) ) ) ) )
=> ( A = B ) ) ) ) ) ).
% i.add.r_cancel
thf(fact_159_i_Oadd_Om__lcomm,axiom,
! [X: int,Y: int,Z: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( member_int @ Y @ ( 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 ) ) ) ) )
=> ( ( member_int @ Z @ ( 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 ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ Z ) )
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ Z ) ) ) ) ) ) ).
% i.add.m_lcomm
thf(fact_160_i_Oadd_Or__inv__ex,axiom,
! [X: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ ( 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 ) ) ) ) )
& ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ X4 )
= ( 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 ) ) ) ) ) ) ) ).
% i.add.r_inv_ex
thf(fact_161_i_Oadd_Oone__unique,axiom,
! [U: int] :
( ( member_int @ U @ ( 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 ) ) ) ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ ( 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 ) ) ) ) )
=> ( ( add_int_Product_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 ) ) ) @ U @ X4 )
= X4 ) )
=> ( U
= ( 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 ) ) ) ) ) ) ) ).
% i.add.one_unique
thf(fact_162_i_Oadd_Ol__inv__ex,axiom,
! [X: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ ( 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 ) ) ) ) )
& ( ( add_int_Product_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 ) ) ) @ X4 @ X )
= ( 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 ) ) ) ) ) ) ) ).
% i.add.l_inv_ex
thf(fact_163_i_Oadd_Oinv__unique,axiom,
! [Y: int,X: int,Y2: int] :
( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ X )
= ( 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 ) ) ) ) )
=> ( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ Y2 )
= ( 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 ) ) ) ) )
=> ( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( member_int @ Y @ ( 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 ) ) ) ) )
=> ( ( member_int @ Y2 @ ( 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 ) ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% i.add.inv_unique
thf(fact_164_i_Oadd_Oinv__comm,axiom,
! [X: int,Y: int] :
( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ Y )
= ( 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 ) ) ) ) )
=> ( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( member_int @ Y @ ( 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 ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ X )
= ( 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 ) ) ) ) ) ) ) ) ).
% i.add.inv_comm
thf(fact_165_mult__minus1__right,axiom,
! [Z: int] :
( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1_right
thf(fact_166_mult__minus1,axiom,
! [Z: int] :
( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
= ( uminus_uminus_int @ Z ) ) ).
% mult_minus1
thf(fact_167_i_Oa__rcos__sum,axiom,
! [A: int,B: int] :
( ( member_int @ A @ top_top_set_int )
=> ( ( member_int @ B @ 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 ) ) ) @ ( 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 ) ) ) @ i @ A ) @ ( 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 ) ) ) @ i @ B ) )
= ( 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 ) ) ) @ i @ ( plus_plus_int @ A @ B ) ) ) ) ) ).
% i.a_rcos_sum
thf(fact_168_i_Orcosets__add__eq,axiom,
! [M: set_int] :
( ( member_set_int @ M @ ( 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 ) ) ) @ i ) )
=> ( ( 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 ) ) ) @ i @ M )
= M ) ) ).
% i.rcosets_add_eq
thf(fact_169_i_Oa__setmult__closed,axiom,
! [K1: set_int,K22: set_int] :
( ( member_set_int @ K1 @ ( 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 ) ) ) @ i ) )
=> ( ( member_set_int @ K22 @ ( 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 ) ) ) @ i ) )
=> ( member_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 ) ) ) @ K1 @ K22 ) @ ( 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 ) ) ) @ i ) ) ) ) ).
% i.a_setmult_closed
thf(fact_170_i_Oa__rcosets__assoc,axiom,
! [M1: set_int,M2: set_int,M3: set_int] :
( ( member_set_int @ M1 @ ( 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 ) ) ) @ i ) )
=> ( ( member_set_int @ M2 @ ( 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 ) ) ) @ i ) )
=> ( ( member_set_int @ M3 @ ( 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 ) ) ) @ i ) )
=> ( ( 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 ) ) ) @ ( 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 ) ) ) @ M1 @ M2 ) @ M3 )
= ( 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 ) ) ) @ M1 @ ( 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 ) ) ) @ M2 @ M3 ) ) ) ) ) ) ).
% i.a_rcosets_assoc
thf(fact_171_i_Oadd_Oright__cancel,axiom,
! [X: int,Y: int,Z: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( member_int @ Y @ ( 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 ) ) ) ) )
=> ( ( member_int @ Z @ ( 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 ) ) ) ) )
=> ( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ X )
= ( add_int_Product_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 ) ) ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% i.add.right_cancel
thf(fact_172_i_Oadd_Om__closed,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( member_int @ Y @ ( 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 ) ) ) ) )
=> ( member_int @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ Y ) @ ( 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 ) ) ) ) ) ) ) ).
% i.add.m_closed
thf(fact_173_i_Oadd_Oone__closed,axiom,
member_int @ ( 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 ) ) ) ) @ ( 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 ) ) ) ) ).
% i.add.one_closed
thf(fact_174_i_Oa__closed,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ i )
=> ( ( member_int @ Y @ i )
=> ( member_int @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ Y ) @ i ) ) ) ).
% i.a_closed
thf(fact_175_i_Oadd_Or__one,axiom,
! [X: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ ( 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 ) ) ) ) )
= X ) ) ).
% i.add.r_one
thf(fact_176_i_Oadd_Or__cancel__one_H,axiom,
! [X: int,A: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( member_int @ A @ ( 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 ) ) ) ) )
=> ( ( X
= ( add_int_Product_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 @ X ) )
= ( A
= ( 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 ) ) ) ) ) ) ) ) ).
% i.add.r_cancel_one'
thf(fact_177_i_Oadd_Or__cancel__one,axiom,
! [X: int,A: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( member_int @ A @ ( 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 ) ) ) ) )
=> ( ( ( add_int_Product_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 @ X )
= X )
= ( A
= ( 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 ) ) ) ) ) ) ) ) ).
% i.add.r_cancel_one
thf(fact_178_i_Oadd_Ol__one,axiom,
! [X: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( add_int_Product_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_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 ) ) ) ) @ X )
= X ) ) ).
% i.add.l_one
thf(fact_179_i_Oadd_Ol__cancel__one_H,axiom,
! [X: int,A: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( member_int @ A @ ( 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 ) ) ) ) )
=> ( ( X
= ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ A ) )
= ( A
= ( 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 ) ) ) ) ) ) ) ) ).
% i.add.l_cancel_one'
thf(fact_180_i_Oadd_Ol__cancel__one,axiom,
! [X: int,A: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( member_int @ A @ ( 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 ) ) ) ) )
=> ( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ A )
= X )
= ( A
= ( 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 ) ) ) ) ) ) ) ) ).
% i.add.l_cancel_one
thf(fact_181_int_Oadd_Ocarrier__not__empty,axiom,
top_top_set_int != bot_bot_set_int ).
% int.add.carrier_not_empty
thf(fact_182_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_183_int_Oadd_Oone__in__subset,axiom,
! [H2: set_int] :
( ( ord_less_eq_set_int @ H2 @ top_top_set_int )
=> ( ( H2 != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ H2 )
=> ( member_int @ ( uminus_uminus_int @ X4 ) @ H2 ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ H2 )
=> ! [Xa: int] :
( ( member_int @ Xa @ H2 )
=> ( member_int @ ( plus_plus_int @ X4 @ Xa ) @ H2 ) ) )
=> ( member_int @ zero_zero_int @ H2 ) ) ) ) ) ).
% int.add.one_in_subset
thf(fact_184_int_Ocanonical__proj__vimage__in__carrier,axiom,
! [I2: set_int,J: 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 @ J @ ( 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 @ J ) @ top_top_set_int ) ) ) ).
% int.canonical_proj_vimage_in_carrier
thf(fact_185_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_186_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_187_int_Ocanonical__proj__vimage__mem__iff,axiom,
! [I2: set_int,J: 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 @ J @ ( 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 @ J ) )
= ( 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 ) @ J ) ) ) ) ) ).
% int.canonical_proj_vimage_mem_iff
thf(fact_188_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_189_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_190_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_191_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_192_int__add__eq,axiom,
! [X: int,Y: int] :
( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ Y )
= ( plus_plus_int @ X @ Y ) ) ).
% int_add_eq
thf(fact_193_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_194_int_Osetadd__subset__G,axiom,
! [H2: set_int,K: set_int] :
( ( ord_less_eq_set_int @ H2 @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ K @ 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 ) ) ) @ H2 @ K ) @ top_top_set_int ) ) ) ).
% int.setadd_subset_G
thf(fact_195_int_Oset__add__closed,axiom,
! [A3: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A3 @ 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 ) ) ) @ A3 @ B2 ) @ top_top_set_int ) ) ) ).
% int.set_add_closed
thf(fact_196_int_Oset__add__comm,axiom,
! [I2: set_int,J: set_int] :
( ( ord_less_eq_set_int @ I2 @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ J @ 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 @ J )
= ( 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 ) ) ) @ J @ I2 ) ) ) ) ).
% int.set_add_comm
thf(fact_197_int_Oadd__ideals,axiom,
! [I2: set_int,J: 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 @ J @ ( 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 @ J ) @ ( 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_198_int_Oadd__additive__subgroups,axiom,
! [H2: set_int,K: set_int] :
( ( additi1413919225584036690t_unit @ H2 @ ( 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 @ K @ ( 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 ) ) ) @ H2 @ K ) @ ( 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_199_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_200_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_201_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_202_int_Oa__setmult__rcos__assoc,axiom,
! [H2: set_int,K: set_int,X: int] :
( ( ord_less_eq_set_int @ H2 @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ K @ 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 ) ) ) @ 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 ) ) ) @ K @ 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 ) ) ) @ H2 @ K ) @ X ) ) ) ) ) ).
% int.a_setmult_rcos_assoc
thf(fact_203_int_Oa__rcos__assoc__lcos,axiom,
! [H2: set_int,K: set_int,X: int] :
( ( ord_less_eq_set_int @ H2 @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ K @ 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 ) ) ) @ ( 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 ) ) ) @ H2 @ X ) @ K )
= ( 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 ) ) ) @ H2 @ ( a_l_co457318074463736598t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ K ) ) ) ) ) ) ).
% int.a_rcos_assoc_lcos
thf(fact_204_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_205_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_206_ring_Oint__embed__ring__hom,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ring_h4095663460228855494t_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 ) ) ) @ R2 @ ( ring_i2743490682209504680t_unit @ R2 ) ) ) ).
% ring.int_embed_ring_hom
thf(fact_207_ring_Oint__embed__ring__hom,axiom,
! [R2: partia2818514838349642498t_unit] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( ring_h1606527851073305852t_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 ) ) ) @ R2 @ ( ring_i2003929451149106142t_unit @ R2 ) ) ) ).
% ring.int_embed_ring_hom
thf(fact_208_s_Oset__add__ker__hom_I1_J,axiom,
! [I2: set_int] :
( ( ord_less_eq_set_int @ I2 @ top_top_set_int )
=> ( ( 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 ) ) ) @ i ) @ ( 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 @ ( a_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) ) ) )
= ( 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 ) ) ) @ i ) @ I2 ) ) ) ).
% s.set_add_ker_hom(1)
thf(fact_209_s_Oset__add__ker__hom_I2_J,axiom,
! [I2: set_int] :
( ( ord_less_eq_set_int @ I2 @ top_top_set_int )
=> ( ( 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 ) ) ) @ i ) @ ( 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 ) ) ) @ ( a_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) ) @ I2 ) )
= ( 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 ) ) ) @ i ) @ I2 ) ) ) ).
% s.set_add_ker_hom(2)
thf(fact_210_i_Oadd_Oone__in__subset,axiom,
! [H2: set_int] :
( ( ord_less_eq_set_int @ H2 @ ( 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 ) ) ) ) )
=> ( ( H2 != bot_bot_set_int )
=> ( ! [X4: int] :
( ( member_int @ X4 @ H2 )
=> ( member_int @ ( a_inv_8811962894454695315t_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 ) ) ) @ X4 ) @ H2 ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ H2 )
=> ! [Xa: int] :
( ( member_int @ Xa @ H2 )
=> ( member_int @ ( add_int_Product_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 ) ) ) @ X4 @ Xa ) @ H2 ) ) )
=> ( member_int @ ( 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 ) ) ) ) @ H2 ) ) ) ) ) ).
% i.add.one_in_subset
thf(fact_211_i_Oadd_Oinv__equality,axiom,
! [Y: int,X: int] :
( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ X )
= ( 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 ) ) ) ) )
=> ( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( member_int @ Y @ ( 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 ) ) ) ) )
=> ( ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X )
= Y ) ) ) ) ).
% i.add.inv_equality
thf(fact_212_Sup__empty,axiom,
( ( comple3221217463730067765et_int @ bot_bot_set_set_int )
= bot_bot_set_int ) ).
% Sup_empty
thf(fact_213_Sup__empty,axiom,
( ( comple7281953568134767595et_int @ bot_bo2384636101374064866et_int )
= bot_bot_set_set_int ) ).
% Sup_empty
thf(fact_214_boolean__algebra_Ocompl__one,axiom,
( ( uminus7346710233107665121et_int @ top_top_set_set_int )
= bot_bot_set_set_int ) ).
% boolean_algebra.compl_one
thf(fact_215_boolean__algebra_Ocompl__one,axiom,
( ( uminus1532241313380277803et_int @ top_top_set_int )
= bot_bot_set_int ) ).
% boolean_algebra.compl_one
thf(fact_216_boolean__algebra_Ocompl__one,axiom,
( ( uminus5710092332889474511et_nat @ top_top_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.compl_one
thf(fact_217_boolean__algebra_Ocompl__zero,axiom,
( ( uminus7346710233107665121et_int @ bot_bot_set_set_int )
= top_top_set_set_int ) ).
% boolean_algebra.compl_zero
thf(fact_218_boolean__algebra_Ocompl__zero,axiom,
( ( uminus1532241313380277803et_int @ bot_bot_set_int )
= top_top_set_int ) ).
% boolean_algebra.compl_zero
thf(fact_219_boolean__algebra_Ocompl__zero,axiom,
( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.compl_zero
thf(fact_220_Sup__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Sup_UNIV
thf(fact_221_Sup__UNIV,axiom,
( ( comple3221217463730067765et_int @ top_top_set_set_int )
= top_top_set_int ) ).
% Sup_UNIV
thf(fact_222_Sup__UNIV,axiom,
( ( comple7281953568134767595et_int @ top_to5524576366173240574et_int )
= top_top_set_set_int ) ).
% Sup_UNIV
thf(fact_223_n__ge__0,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% n_ge_0
thf(fact_224_r_Ocarrier__not__empty,axiom,
( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
!= bot_bot_set_set_int ) ).
% r.carrier_not_empty
thf(fact_225_int_Ole__refl,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ord_less_eq_int @ X @ X ) ) ).
% int.le_refl
thf(fact_226_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_227_Union__iff,axiom,
! [A3: int > set_int,C2: set_set_int_set_int] :
( ( member_int_set_int @ A3 @ ( comple8192055500899288118et_int @ C2 ) )
= ( ? [X5: set_int_set_int] :
( ( member7905727001773941114et_int @ X5 @ C2 )
& ( member_int_set_int @ A3 @ X5 ) ) ) ) ).
% Union_iff
thf(fact_228_Union__iff,axiom,
! [A3: set_set_int,C2: set_set_set_set_int] :
( ( member_set_set_int @ A3 @ ( comple1756060948637632417et_int @ C2 ) )
= ( ? [X5: set_set_set_int] :
( ( member7356822600254261989et_int @ X5 @ C2 )
& ( member_set_set_int @ A3 @ X5 ) ) ) ) ).
% Union_iff
thf(fact_229_Union__iff,axiom,
! [A3: nat,C2: set_set_nat] :
( ( member_nat @ A3 @ ( comple7399068483239264473et_nat @ C2 ) )
= ( ? [X5: set_nat] :
( ( member_set_nat @ X5 @ C2 )
& ( member_nat @ A3 @ X5 ) ) ) ) ).
% Union_iff
thf(fact_230_Union__iff,axiom,
! [A3: int,C2: set_set_int] :
( ( member_int @ A3 @ ( comple3221217463730067765et_int @ C2 ) )
= ( ? [X5: set_int] :
( ( member_set_int @ X5 @ C2 )
& ( member_int @ A3 @ X5 ) ) ) ) ).
% Union_iff
thf(fact_231_Union__iff,axiom,
! [A3: set_int,C2: set_set_set_int] :
( ( member_set_int @ A3 @ ( comple7281953568134767595et_int @ C2 ) )
= ( ? [X5: set_set_int] :
( ( member_set_set_int @ X5 @ C2 )
& ( member_set_int @ A3 @ X5 ) ) ) ) ).
% Union_iff
thf(fact_232_UnionI,axiom,
! [X3: set_int_set_int,C2: set_set_int_set_int,A3: int > set_int] :
( ( member7905727001773941114et_int @ X3 @ C2 )
=> ( ( member_int_set_int @ A3 @ X3 )
=> ( member_int_set_int @ A3 @ ( comple8192055500899288118et_int @ C2 ) ) ) ) ).
% UnionI
thf(fact_233_UnionI,axiom,
! [X3: set_set_set_int,C2: set_set_set_set_int,A3: set_set_int] :
( ( member7356822600254261989et_int @ X3 @ C2 )
=> ( ( member_set_set_int @ A3 @ X3 )
=> ( member_set_set_int @ A3 @ ( comple1756060948637632417et_int @ C2 ) ) ) ) ).
% UnionI
thf(fact_234_UnionI,axiom,
! [X3: set_nat,C2: set_set_nat,A3: nat] :
( ( member_set_nat @ X3 @ C2 )
=> ( ( member_nat @ A3 @ X3 )
=> ( member_nat @ A3 @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_235_UnionI,axiom,
! [X3: set_int,C2: set_set_int,A3: int] :
( ( member_set_int @ X3 @ C2 )
=> ( ( member_int @ A3 @ X3 )
=> ( member_int @ A3 @ ( comple3221217463730067765et_int @ C2 ) ) ) ) ).
% UnionI
thf(fact_236_UnionI,axiom,
! [X3: set_set_int,C2: set_set_set_int,A3: set_int] :
( ( member_set_set_int @ X3 @ C2 )
=> ( ( member_set_int @ A3 @ X3 )
=> ( member_set_int @ A3 @ ( comple7281953568134767595et_int @ C2 ) ) ) ) ).
% UnionI
thf(fact_237_UN__ball__bex__simps_I1_J,axiom,
! [A3: set_set_int,P: int > $o] :
( ( ! [X5: int] :
( ( member_int @ X5 @ ( comple3221217463730067765et_int @ A3 ) )
=> ( P @ X5 ) ) )
= ( ! [X5: set_int] :
( ( member_set_int @ X5 @ A3 )
=> ! [Y3: int] :
( ( member_int @ Y3 @ X5 )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_238_UN__ball__bex__simps_I1_J,axiom,
! [A3: set_set_set_int,P: set_int > $o] :
( ( ! [X5: set_int] :
( ( member_set_int @ X5 @ ( comple7281953568134767595et_int @ A3 ) )
=> ( P @ X5 ) ) )
= ( ! [X5: set_set_int] :
( ( member_set_set_int @ X5 @ A3 )
=> ! [Y3: set_int] :
( ( member_set_int @ Y3 @ X5 )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_239_UN__ball__bex__simps_I3_J,axiom,
! [A3: set_set_int,P: int > $o] :
( ( ? [X5: int] :
( ( member_int @ X5 @ ( comple3221217463730067765et_int @ A3 ) )
& ( P @ X5 ) ) )
= ( ? [X5: set_int] :
( ( member_set_int @ X5 @ A3 )
& ? [Y3: int] :
( ( member_int @ Y3 @ X5 )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_240_UN__ball__bex__simps_I3_J,axiom,
! [A3: set_set_set_int,P: set_int > $o] :
( ( ? [X5: set_int] :
( ( member_set_int @ X5 @ ( comple7281953568134767595et_int @ A3 ) )
& ( P @ X5 ) ) )
= ( ? [X5: set_set_int] :
( ( member_set_set_int @ X5 @ A3 )
& ? [Y3: set_int] :
( ( member_set_int @ Y3 @ X5 )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_241_compl__le__compl__iff,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y ) )
= ( ord_less_eq_set_int @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_242_compl__le__compl__iff,axiom,
! [X: set_set_int,Y: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( uminus7346710233107665121et_int @ X ) @ ( uminus7346710233107665121et_int @ Y ) )
= ( ord_le4403425263959731960et_int @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_243_compl__le__compl__iff,axiom,
! [X: set_set_set_int,Y: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ ( uminus3047225438891465879et_int @ X ) @ ( uminus3047225438891465879et_int @ Y ) )
= ( ord_le4317611570275147438et_int @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_244_Sup__bot__conv_I1_J,axiom,
! [A3: set_set_int] :
( ( ( comple3221217463730067765et_int @ A3 )
= bot_bot_set_int )
= ( ! [X5: set_int] :
( ( member_set_int @ X5 @ A3 )
=> ( X5 = bot_bot_set_int ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_245_Sup__bot__conv_I1_J,axiom,
! [A3: set_set_set_int] :
( ( ( comple7281953568134767595et_int @ A3 )
= bot_bot_set_set_int )
= ( ! [X5: set_set_int] :
( ( member_set_set_int @ X5 @ A3 )
=> ( X5 = bot_bot_set_set_int ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_246_Sup__bot__conv_I2_J,axiom,
! [A3: set_set_int] :
( ( bot_bot_set_int
= ( comple3221217463730067765et_int @ A3 ) )
= ( ! [X5: set_int] :
( ( member_set_int @ X5 @ A3 )
=> ( X5 = bot_bot_set_int ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_247_Sup__bot__conv_I2_J,axiom,
! [A3: set_set_set_int] :
( ( bot_bot_set_set_int
= ( comple7281953568134767595et_int @ A3 ) )
= ( ! [X5: set_set_int] :
( ( member_set_set_int @ X5 @ A3 )
=> ( X5 = bot_bot_set_set_int ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_248_ball__UN,axiom,
! [B2: int > set_int,A3: set_int,P: int > $o] :
( ( ! [X5: int] :
( ( member_int @ X5 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B2 @ A3 ) ) )
=> ( P @ X5 ) ) )
= ( ! [X5: int] :
( ( member_int @ X5 @ A3 )
=> ! [Y3: int] :
( ( member_int @ Y3 @ ( B2 @ X5 ) )
=> ( P @ Y3 ) ) ) ) ) ).
% ball_UN
thf(fact_249_ball__UN,axiom,
! [B2: set_int > set_set_int,A3: set_set_int,P: set_int > $o] :
( ( ! [X5: set_int] :
( ( member_set_int @ X5 @ ( comple7281953568134767595et_int @ ( image_1010086626112315521et_int @ B2 @ A3 ) ) )
=> ( P @ X5 ) ) )
= ( ! [X5: set_int] :
( ( member_set_int @ X5 @ A3 )
=> ! [Y3: set_int] :
( ( member_set_int @ Y3 @ ( B2 @ X5 ) )
=> ( P @ Y3 ) ) ) ) ) ).
% ball_UN
thf(fact_250_bex__UN,axiom,
! [B2: int > set_int,A3: set_int,P: int > $o] :
( ( ? [X5: int] :
( ( member_int @ X5 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B2 @ A3 ) ) )
& ( P @ X5 ) ) )
= ( ? [X5: int] :
( ( member_int @ X5 @ A3 )
& ? [Y3: int] :
( ( member_int @ Y3 @ ( B2 @ X5 ) )
& ( P @ Y3 ) ) ) ) ) ).
% bex_UN
thf(fact_251_bex__UN,axiom,
! [B2: set_int > set_set_int,A3: set_set_int,P: set_int > $o] :
( ( ? [X5: set_int] :
( ( member_set_int @ X5 @ ( comple7281953568134767595et_int @ ( image_1010086626112315521et_int @ B2 @ A3 ) ) )
& ( P @ X5 ) ) )
= ( ? [X5: set_int] :
( ( member_set_int @ X5 @ A3 )
& ? [Y3: set_int] :
( ( member_set_int @ Y3 @ ( B2 @ X5 ) )
& ( P @ Y3 ) ) ) ) ) ).
% bex_UN
thf(fact_252_UN__ball__bex__simps_I2_J,axiom,
! [B2: int > set_int,A3: set_int,P: int > $o] :
( ( ! [X5: int] :
( ( member_int @ X5 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B2 @ A3 ) ) )
=> ( P @ X5 ) ) )
= ( ! [X5: int] :
( ( member_int @ X5 @ A3 )
=> ! [Y3: int] :
( ( member_int @ Y3 @ ( B2 @ X5 ) )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_253_UN__ball__bex__simps_I2_J,axiom,
! [B2: set_int > set_set_int,A3: set_set_int,P: set_int > $o] :
( ( ! [X5: set_int] :
( ( member_set_int @ X5 @ ( comple7281953568134767595et_int @ ( image_1010086626112315521et_int @ B2 @ A3 ) ) )
=> ( P @ X5 ) ) )
= ( ! [X5: set_int] :
( ( member_set_int @ X5 @ A3 )
=> ! [Y3: set_int] :
( ( member_set_int @ Y3 @ ( B2 @ X5 ) )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_254_UN__ball__bex__simps_I4_J,axiom,
! [B2: int > set_int,A3: set_int,P: int > $o] :
( ( ? [X5: int] :
( ( member_int @ X5 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B2 @ A3 ) ) )
& ( P @ X5 ) ) )
= ( ? [X5: int] :
( ( member_int @ X5 @ A3 )
& ? [Y3: int] :
( ( member_int @ Y3 @ ( B2 @ X5 ) )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_255_UN__ball__bex__simps_I4_J,axiom,
! [B2: set_int > set_set_int,A3: set_set_int,P: set_int > $o] :
( ( ? [X5: set_int] :
( ( member_set_int @ X5 @ ( comple7281953568134767595et_int @ ( image_1010086626112315521et_int @ B2 @ A3 ) ) )
& ( P @ X5 ) ) )
= ( ? [X5: set_int] :
( ( member_set_int @ X5 @ A3 )
& ? [Y3: set_int] :
( ( member_set_int @ Y3 @ ( B2 @ X5 ) )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_256_i_Oadd_Oinv__mult,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( member_int @ Y @ ( 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 ) ) ) ) )
=> ( ( a_inv_8811962894454695315t_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 ) ) ) @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ Y ) )
= ( add_int_Product_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_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X ) @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y ) ) ) ) ) ).
% i.add.inv_mult
thf(fact_257_i_Oadd_Oinv__mult__group,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( member_int @ Y @ ( 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 ) ) ) ) )
=> ( ( a_inv_8811962894454695315t_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 ) ) ) @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ Y ) )
= ( add_int_Product_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_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y ) @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X ) ) ) ) ) ).
% i.add.inv_mult_group
thf(fact_258_i_Oadd_Oinv__solve__left,axiom,
! [A: int,B: int,C: int] :
( ( member_int @ A @ ( 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 ) ) ) ) )
=> ( ( member_int @ B @ ( 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 ) ) ) ) )
=> ( ( member_int @ C @ ( 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 ) ) ) ) )
=> ( ( A
= ( add_int_Product_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_inv_8811962894454695315t_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 ) ) ) @ B ) @ C ) )
= ( C
= ( add_int_Product_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 ) ) ) @ B @ A ) ) ) ) ) ) ).
% i.add.inv_solve_left
thf(fact_259_i_Oadd_Oinv__solve__left_H,axiom,
! [A: int,B: int,C: int] :
( ( member_int @ A @ ( 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 ) ) ) ) )
=> ( ( member_int @ B @ ( 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 ) ) ) ) )
=> ( ( member_int @ C @ ( 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 ) ) ) ) )
=> ( ( ( add_int_Product_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_inv_8811962894454695315t_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 ) ) ) @ B ) @ C )
= A )
= ( C
= ( add_int_Product_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 ) ) ) @ B @ A ) ) ) ) ) ) ).
% i.add.inv_solve_left'
thf(fact_260_i_Oadd_Oinv__solve__right,axiom,
! [A: int,B: int,C: int] :
( ( member_int @ A @ ( 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 ) ) ) ) )
=> ( ( member_int @ B @ ( 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 ) ) ) ) )
=> ( ( member_int @ C @ ( 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 ) ) ) ) )
=> ( ( A
= ( add_int_Product_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 ) ) ) @ B @ ( a_inv_8811962894454695315t_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 ) ) ) @ C ) ) )
= ( B
= ( add_int_Product_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 @ C ) ) ) ) ) ) ).
% i.add.inv_solve_right
thf(fact_261_i_Oadd_Oinv__solve__right_H,axiom,
! [A: int,B: int,C: int] :
( ( member_int @ A @ ( 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 ) ) ) ) )
=> ( ( member_int @ B @ ( 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 ) ) ) ) )
=> ( ( member_int @ C @ ( 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 ) ) ) ) )
=> ( ( ( add_int_Product_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 ) ) ) @ B @ ( a_inv_8811962894454695315t_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 ) ) ) @ C ) )
= A )
= ( B
= ( add_int_Product_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 @ C ) ) ) ) ) ) ).
% i.add.inv_solve_right'
thf(fact_262_i_Oadd_Oinv__closed,axiom,
! [X: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( member_int @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X ) @ ( 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 ) ) ) ) ) ) ).
% i.add.inv_closed
thf(fact_263_i_Oadd_Oinv__inv,axiom,
! [X: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( a_inv_8811962894454695315t_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_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X ) )
= X ) ) ).
% i.add.inv_inv
thf(fact_264_i_Oa__inv__closed,axiom,
! [X: int] :
( ( member_int @ X @ i )
=> ( member_int @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X ) @ i ) ) ).
% i.a_inv_closed
thf(fact_265_i_Oadd_Oinv__eq__1__iff,axiom,
! [X: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X )
= ( 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 ) ) ) ) )
= ( X
= ( 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 ) ) ) ) ) ) ) ).
% i.add.inv_eq_1_iff
thf(fact_266_i_Oadd_Ol__inv,axiom,
! [X: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( add_int_Product_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_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X ) @ X )
= ( 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 ) ) ) ) ) ) ).
% i.add.l_inv
thf(fact_267_i_Oadd_Or__inv,axiom,
! [X: int] :
( ( member_int @ X @ ( 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 ) ) ) ) )
=> ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X ) )
= ( 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 ) ) ) ) ) ) ).
% i.add.r_inv
thf(fact_268_SUP__cong,axiom,
! [A3: set_int,B2: set_int,C2: int > set_int,D: int > set_int] :
( ( A3 = B2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_int_set_int @ C2 @ A3 ) )
= ( comple3221217463730067765et_int @ ( image_int_set_int @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_269_SUP__cong,axiom,
! [A3: set_set_int,B2: set_set_int,C2: set_int > set_int,D: set_int > set_int] :
( ( A3 = B2 )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ C2 @ A3 ) )
= ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_270_SUP__cong,axiom,
! [A3: set_int_set_int,B2: set_int_set_int,C2: ( int > set_int ) > set_int,D: ( int > set_int ) > set_int] :
( ( A3 = B2 )
=> ( ! [X4: int > set_int] :
( ( member_int_set_int @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_5113617136271507350et_int @ C2 @ A3 ) )
= ( comple3221217463730067765et_int @ ( image_5113617136271507350et_int @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_271_SUP__cong,axiom,
! [A3: set_set_set_int,B2: set_set_set_int,C2: set_set_int > set_int,D: set_set_int > set_int] :
( ( A3 = B2 )
=> ( ! [X4: set_set_int] :
( ( member_set_set_int @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_3513010637850279041et_int @ C2 @ A3 ) )
= ( comple3221217463730067765et_int @ ( image_3513010637850279041et_int @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_272_SUP__cong,axiom,
! [A3: set_nat,B2: set_nat,C2: nat > set_int,D: nat > set_int] :
( ( A3 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ C2 @ A3 ) )
= ( comple3221217463730067765et_int @ ( image_nat_set_int @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_273_SUP__cong,axiom,
! [A3: set_int,B2: set_int,C2: int > set_set_int,D: int > set_set_int] :
( ( A3 = B2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ C2 @ A3 ) )
= ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_274_SUP__cong,axiom,
! [A3: set_set_int,B2: set_set_int,C2: set_int > set_set_int,D: set_int > set_set_int] :
( ( A3 = B2 )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7281953568134767595et_int @ ( image_1010086626112315521et_int @ C2 @ A3 ) )
= ( comple7281953568134767595et_int @ ( image_1010086626112315521et_int @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_275_SUP__cong,axiom,
! [A3: set_int_set_int,B2: set_int_set_int,C2: ( int > set_int ) > set_set_int,D: ( int > set_int ) > set_set_int] :
( ( A3 = B2 )
=> ( ! [X4: int > set_int] :
( ( member_int_set_int @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7281953568134767595et_int @ ( image_3533601879074506316et_int @ C2 @ A3 ) )
= ( comple7281953568134767595et_int @ ( image_3533601879074506316et_int @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_276_SUP__cong,axiom,
! [A3: set_set_set_int,B2: set_set_set_int,C2: set_set_int > set_set_int,D: set_set_int > set_set_int] :
( ( A3 = B2 )
=> ( ! [X4: set_set_int] :
( ( member_set_set_int @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7281953568134767595et_int @ ( image_2718908958957770551et_int @ C2 @ A3 ) )
= ( comple7281953568134767595et_int @ ( image_2718908958957770551et_int @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_277_SUP__cong,axiom,
! [A3: set_nat,B2: set_nat,C2: nat > set_set_int,D: nat > set_set_int] :
( ( A3 = B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( comple7281953568134767595et_int @ ( image_8927401050382224495et_int @ C2 @ A3 ) )
= ( comple7281953568134767595et_int @ ( image_8927401050382224495et_int @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_278_image__set__eqI,axiom,
! [A3: set_int,F: int > int,B2: set_int,G: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A3 )
=> ( member_int @ ( F @ X4 ) @ B2 ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ B2 )
=> ( ( member_int @ ( G @ X4 ) @ A3 )
& ( ( F @ ( G @ X4 ) )
= X4 ) ) )
=> ( ( image_int_int @ F @ A3 )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_279_image__set__eqI,axiom,
! [A3: set_int,F: int > nat,B2: set_nat,G: nat > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A3 )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( member_int @ ( G @ X4 ) @ A3 )
& ( ( F @ ( G @ X4 ) )
= X4 ) ) )
=> ( ( image_int_nat @ F @ A3 )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_280_image__set__eqI,axiom,
! [A3: set_nat,F: nat > int,B2: set_int,G: int > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_int @ ( F @ X4 ) @ B2 ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ B2 )
=> ( ( member_nat @ ( G @ X4 ) @ A3 )
& ( ( F @ ( G @ X4 ) )
= X4 ) ) )
=> ( ( image_nat_int @ F @ A3 )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_281_image__set__eqI,axiom,
! [A3: set_nat,F: nat > nat,B2: set_nat,G: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( member_nat @ ( G @ X4 ) @ A3 )
& ( ( F @ ( G @ X4 ) )
= X4 ) ) )
=> ( ( image_nat_nat @ F @ A3 )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_282_image__set__eqI,axiom,
! [A3: set_int,F: int > set_int,B2: set_set_int,G: set_int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A3 )
=> ( member_set_int @ ( F @ X4 ) @ B2 ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ B2 )
=> ( ( member_int @ ( G @ X4 ) @ A3 )
& ( ( F @ ( G @ X4 ) )
= X4 ) ) )
=> ( ( image_int_set_int @ F @ A3 )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_283_image__set__eqI,axiom,
! [A3: set_set_int,F: set_int > int,B2: set_int,G: int > set_int] :
( ! [X4: set_int] :
( ( member_set_int @ X4 @ A3 )
=> ( member_int @ ( F @ X4 ) @ B2 ) )
=> ( ! [X4: int] :
( ( member_int @ X4 @ B2 )
=> ( ( member_set_int @ ( G @ X4 ) @ A3 )
& ( ( F @ ( G @ X4 ) )
= X4 ) ) )
=> ( ( image_set_int_int @ F @ A3 )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_284_image__set__eqI,axiom,
! [A3: set_set_int,F: set_int > nat,B2: set_nat,G: nat > set_int] :
( ! [X4: set_int] :
( ( member_set_int @ X4 @ A3 )
=> ( member_nat @ ( F @ X4 ) @ B2 ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( ( member_set_int @ ( G @ X4 ) @ A3 )
& ( ( F @ ( G @ X4 ) )
= X4 ) ) )
=> ( ( image_set_int_nat @ F @ A3 )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_285_image__set__eqI,axiom,
! [A3: set_nat,F: nat > set_int,B2: set_set_int,G: set_int > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_set_int @ ( F @ X4 ) @ B2 ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ B2 )
=> ( ( member_nat @ ( G @ X4 ) @ A3 )
& ( ( F @ ( G @ X4 ) )
= X4 ) ) )
=> ( ( image_nat_set_int @ F @ A3 )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_286_image__set__eqI,axiom,
! [A3: set_int,F: int > set_set_int,B2: set_set_set_int,G: set_set_int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A3 )
=> ( member_set_set_int @ ( F @ X4 ) @ B2 ) )
=> ( ! [X4: set_set_int] :
( ( member_set_set_int @ X4 @ B2 )
=> ( ( member_int @ ( G @ X4 ) @ A3 )
& ( ( F @ ( G @ X4 ) )
= X4 ) ) )
=> ( ( image_6617583118289273547et_int @ F @ A3 )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_287_image__set__eqI,axiom,
! [A3: set_set_int,F: set_int > set_int,B2: set_set_int,G: set_int > set_int] :
( ! [X4: set_int] :
( ( member_set_int @ X4 @ A3 )
=> ( member_set_int @ ( F @ X4 ) @ B2 ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ B2 )
=> ( ( member_set_int @ ( G @ X4 ) @ A3 )
& ( ( F @ ( G @ X4 ) )
= X4 ) ) )
=> ( ( image_524474410958335435et_int @ F @ A3 )
= B2 ) ) ) ).
% image_set_eqI
thf(fact_288_Sup_OSUP__cong,axiom,
! [A3: set_int,B2: set_int,C2: int > set_int,D: int > set_int,Sup: set_set_int > set_int] :
( ( A3 = B2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_int_set_int @ C2 @ A3 ) )
= ( Sup @ ( image_int_set_int @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_289_Sup_OSUP__cong,axiom,
! [A3: set_set_int,B2: set_set_int,C2: set_int > set_set_int,D: set_int > set_set_int,Sup: set_set_set_int > set_set_int] :
( ( A3 = B2 )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_1010086626112315521et_int @ C2 @ A3 ) )
= ( Sup @ ( image_1010086626112315521et_int @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_290_Inf_OINF__cong,axiom,
! [A3: set_int,B2: set_int,C2: int > set_int,D: int > set_int,Inf: set_set_int > set_int] :
( ( A3 = B2 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_int_set_int @ C2 @ A3 ) )
= ( Inf @ ( image_int_set_int @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_291_Inf_OINF__cong,axiom,
! [A3: set_set_int,B2: set_set_int,C2: set_int > set_set_int,D: set_int > set_set_int,Inf: set_set_set_int > set_set_int] :
( ( A3 = B2 )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ B2 )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_1010086626112315521et_int @ C2 @ A3 ) )
= ( Inf @ ( image_1010086626112315521et_int @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_292_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_293_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_294_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_295_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_296_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_297_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_298_SUP__eq,axiom,
! [A3: set_int,B2: set_int,F: int > set_int,G: int > set_int] :
( ! [I3: int] :
( ( member_int @ I3 @ A3 )
=> ? [X2: int] :
( ( member_int @ X2 @ B2 )
& ( ord_less_eq_set_int @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: int] :
( ( member_int @ J2 @ B2 )
=> ? [X2: int] :
( ( member_int @ X2 @ A3 )
& ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) )
= ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_299_SUP__eq,axiom,
! [A3: set_int,B2: set_nat,F: int > set_int,G: nat > set_int] :
( ! [I3: int] :
( ( member_int @ I3 @ A3 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ( ord_less_eq_set_int @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X2: int] :
( ( member_int @ X2 @ A3 )
& ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) )
= ( comple3221217463730067765et_int @ ( image_nat_set_int @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_300_SUP__eq,axiom,
! [A3: set_nat,B2: set_int,F: nat > set_int,G: int > set_int] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A3 )
=> ? [X2: int] :
( ( member_int @ X2 @ B2 )
& ( ord_less_eq_set_int @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: int] :
( ( member_int @ J2 @ B2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) )
= ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_301_SUP__eq,axiom,
! [A3: set_nat,B2: set_nat,F: nat > set_int,G: nat > set_int] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A3 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ( ord_less_eq_set_int @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) )
= ( comple3221217463730067765et_int @ ( image_nat_set_int @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_302_SUP__eq,axiom,
! [A3: set_int,B2: set_set_int,F: int > set_int,G: set_int > set_int] :
( ! [I3: int] :
( ( member_int @ I3 @ A3 )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ B2 )
& ( ord_less_eq_set_int @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: set_int] :
( ( member_set_int @ J2 @ B2 )
=> ? [X2: int] :
( ( member_int @ X2 @ A3 )
& ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) )
= ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_303_SUP__eq,axiom,
! [A3: set_set_int,B2: set_int,F: set_int > set_int,G: int > set_int] :
( ! [I3: set_int] :
( ( member_set_int @ I3 @ A3 )
=> ? [X2: int] :
( ( member_int @ X2 @ B2 )
& ( ord_less_eq_set_int @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: int] :
( ( member_int @ J2 @ B2 )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ A3 )
& ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ A3 ) )
= ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_304_SUP__eq,axiom,
! [A3: set_set_int,B2: set_nat,F: set_int > set_int,G: nat > set_int] :
( ! [I3: set_int] :
( ( member_set_int @ I3 @ A3 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ( ord_less_eq_set_int @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ A3 )
& ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ A3 ) )
= ( comple3221217463730067765et_int @ ( image_nat_set_int @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_305_SUP__eq,axiom,
! [A3: set_nat,B2: set_set_int,F: nat > set_int,G: set_int > set_int] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A3 )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ B2 )
& ( ord_less_eq_set_int @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: set_int] :
( ( member_set_int @ J2 @ B2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( ord_less_eq_set_int @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) )
= ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_306_SUP__eq,axiom,
! [A3: set_int,B2: set_int,F: int > set_set_int,G: int > set_set_int] :
( ! [I3: int] :
( ( member_int @ I3 @ A3 )
=> ? [X2: int] :
( ( member_int @ X2 @ B2 )
& ( ord_le4403425263959731960et_int @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: int] :
( ( member_int @ J2 @ B2 )
=> ? [X2: int] :
( ( member_int @ X2 @ A3 )
& ( ord_le4403425263959731960et_int @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ F @ A3 ) )
= ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_307_SUP__eq,axiom,
! [A3: set_int,B2: set_nat,F: int > set_set_int,G: nat > set_set_int] :
( ! [I3: int] :
( ( member_int @ I3 @ A3 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ( ord_le4403425263959731960et_int @ ( F @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B2 )
=> ? [X2: int] :
( ( member_int @ X2 @ A3 )
& ( ord_le4403425263959731960et_int @ ( G @ J2 ) @ ( F @ X2 ) ) ) )
=> ( ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ F @ A3 ) )
= ( comple7281953568134767595et_int @ ( image_8927401050382224495et_int @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_308_SUP__eq__const,axiom,
! [I2: set_int_set_int,F: ( int > set_int ) > set_int,X: set_int] :
( ( I2 != bot_bo5203049024207062007et_int )
=> ( ! [I3: int > set_int] :
( ( member_int_set_int @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3221217463730067765et_int @ ( image_5113617136271507350et_int @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_309_SUP__eq__const,axiom,
! [I2: set_set_set_int,F: set_set_int > set_int,X: set_int] :
( ( I2 != bot_bo2384636101374064866et_int )
=> ( ! [I3: set_set_int] :
( ( member_set_set_int @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3221217463730067765et_int @ ( image_3513010637850279041et_int @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_310_SUP__eq__const,axiom,
! [I2: set_nat,F: nat > set_int,X: set_int] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_311_SUP__eq__const,axiom,
! [I2: set_int,F: int > set_int,X: set_int] :
( ( I2 != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_312_SUP__eq__const,axiom,
! [I2: set_set_int,F: set_int > set_int,X: set_int] :
( ( I2 != bot_bot_set_set_int )
=> ( ! [I3: set_int] :
( ( member_set_int @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_313_SUP__eq__const,axiom,
! [I2: set_int_set_int,F: ( int > set_int ) > set_set_int,X: set_set_int] :
( ( I2 != bot_bo5203049024207062007et_int )
=> ( ! [I3: int > set_int] :
( ( member_int_set_int @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple7281953568134767595et_int @ ( image_3533601879074506316et_int @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_314_SUP__eq__const,axiom,
! [I2: set_set_set_int,F: set_set_int > set_set_int,X: set_set_int] :
( ( I2 != bot_bo2384636101374064866et_int )
=> ( ! [I3: set_set_int] :
( ( member_set_set_int @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple7281953568134767595et_int @ ( image_2718908958957770551et_int @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_315_SUP__eq__const,axiom,
! [I2: set_nat,F: nat > set_set_int,X: set_set_int] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple7281953568134767595et_int @ ( image_8927401050382224495et_int @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_316_SUP__eq__const,axiom,
! [I2: set_int,F: int > set_set_int,X: set_set_int] :
( ( I2 != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_317_SUP__eq__const,axiom,
! [I2: set_set_int,F: set_int > set_set_int,X: set_set_int] :
( ( I2 != bot_bot_set_set_int )
=> ( ! [I3: set_int] :
( ( member_set_int @ I3 @ I2 )
=> ( ( F @ I3 )
= X ) )
=> ( ( comple7281953568134767595et_int @ ( image_1010086626112315521et_int @ F @ I2 ) )
= X ) ) ) ).
% SUP_eq_const
thf(fact_318_SUP__eq__iff,axiom,
! [I2: set_nat,C: set_int,F: nat > set_int] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ord_less_eq_set_int @ C @ ( F @ I3 ) ) )
=> ( ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ I2 ) )
= C )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_319_SUP__eq__iff,axiom,
! [I2: set_int,C: set_int,F: int > set_int] :
( ( I2 != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I2 )
=> ( ord_less_eq_set_int @ C @ ( F @ I3 ) ) )
=> ( ( ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ I2 ) )
= C )
= ( ! [X5: int] :
( ( member_int @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_320_SUP__eq__iff,axiom,
! [I2: set_set_int,C: set_int,F: set_int > set_int] :
( ( I2 != bot_bot_set_set_int )
=> ( ! [I3: set_int] :
( ( member_set_int @ I3 @ I2 )
=> ( ord_less_eq_set_int @ C @ ( F @ I3 ) ) )
=> ( ( ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ I2 ) )
= C )
= ( ! [X5: set_int] :
( ( member_set_int @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_321_SUP__eq__iff,axiom,
! [I2: set_nat,C: set_set_int,F: nat > set_set_int] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ord_le4403425263959731960et_int @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7281953568134767595et_int @ ( image_8927401050382224495et_int @ F @ I2 ) )
= C )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_322_SUP__eq__iff,axiom,
! [I2: set_int,C: set_set_int,F: int > set_set_int] :
( ( I2 != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I2 )
=> ( ord_le4403425263959731960et_int @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ F @ I2 ) )
= C )
= ( ! [X5: int] :
( ( member_int @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_323_SUP__eq__iff,axiom,
! [I2: set_nat,C: set_set_set_int,F: nat > set_set_set_int] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I2 )
=> ( ord_le4317611570275147438et_int @ C @ ( F @ I3 ) ) )
=> ( ( ( comple1756060948637632417et_int @ ( image_924495994448699429et_int @ F @ I2 ) )
= C )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_324_SUP__eq__iff,axiom,
! [I2: set_int,C: set_set_set_int,F: int > set_set_set_int] :
( ( I2 != bot_bot_set_int )
=> ( ! [I3: int] :
( ( member_int @ I3 @ I2 )
=> ( ord_le4317611570275147438et_int @ C @ ( F @ I3 ) ) )
=> ( ( ( comple1756060948637632417et_int @ ( image_63215931098211969et_int @ F @ I2 ) )
= C )
= ( ! [X5: int] :
( ( member_int @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_325_SUP__eq__iff,axiom,
! [I2: set_set_set_int,C: set_int,F: set_set_int > set_int] :
( ( I2 != bot_bo2384636101374064866et_int )
=> ( ! [I3: set_set_int] :
( ( member_set_set_int @ I3 @ I2 )
=> ( ord_less_eq_set_int @ C @ ( F @ I3 ) ) )
=> ( ( ( comple3221217463730067765et_int @ ( image_3513010637850279041et_int @ F @ I2 ) )
= C )
= ( ! [X5: set_set_int] :
( ( member_set_set_int @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_326_SUP__eq__iff,axiom,
! [I2: set_set_int,C: set_set_int,F: set_int > set_set_int] :
( ( I2 != bot_bot_set_set_int )
=> ( ! [I3: set_int] :
( ( member_set_int @ I3 @ I2 )
=> ( ord_le4403425263959731960et_int @ C @ ( F @ I3 ) ) )
=> ( ( ( comple7281953568134767595et_int @ ( image_1010086626112315521et_int @ F @ I2 ) )
= C )
= ( ! [X5: set_int] :
( ( member_set_int @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_327_SUP__eq__iff,axiom,
! [I2: set_set_int,C: set_set_set_int,F: set_int > set_set_set_int] :
( ( I2 != bot_bot_set_set_int )
=> ( ! [I3: set_int] :
( ( member_set_int @ I3 @ I2 )
=> ( ord_le4317611570275147438et_int @ C @ ( F @ I3 ) ) )
=> ( ( ( comple1756060948637632417et_int @ ( image_1737623724905622583et_int @ F @ I2 ) )
= C )
= ( ! [X5: set_int] :
( ( member_set_int @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_328_ring_Oint__embed__inv,axiom,
! [R2: partia4934656038542163276t_unit,X: int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ring_i2743490682209504680t_unit @ R2 @ ( uminus_uminus_int @ X ) )
= ( a_inv_5951419416477254493t_unit @ R2 @ ( ring_i2743490682209504680t_unit @ R2 @ X ) ) ) ) ).
% ring.int_embed_inv
thf(fact_329_ring_Oint__embed__inv,axiom,
! [R2: partia2818514838349642498t_unit,X: int] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( ( ring_i2003929451149106142t_unit @ R2 @ ( uminus_uminus_int @ X ) )
= ( a_inv_8811962894454695315t_unit @ R2 @ ( ring_i2003929451149106142t_unit @ R2 @ X ) ) ) ) ).
% ring.int_embed_inv
thf(fact_330_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_331_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_332_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_333_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_334_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_335_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_336_int_Oint__embed__ring__hom,axiom,
ring_h1606527851073305852t_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 ) ) ) @ ( 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 ) ) ) @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ).
% int.int_embed_ring_hom
thf(fact_337_UnionE,axiom,
! [A3: int > set_int,C2: set_set_int_set_int] :
( ( member_int_set_int @ A3 @ ( comple8192055500899288118et_int @ C2 ) )
=> ~ ! [X8: set_int_set_int] :
( ( member_int_set_int @ A3 @ X8 )
=> ~ ( member7905727001773941114et_int @ X8 @ C2 ) ) ) ).
% UnionE
thf(fact_338_UnionE,axiom,
! [A3: set_set_int,C2: set_set_set_set_int] :
( ( member_set_set_int @ A3 @ ( comple1756060948637632417et_int @ C2 ) )
=> ~ ! [X8: set_set_set_int] :
( ( member_set_set_int @ A3 @ X8 )
=> ~ ( member7356822600254261989et_int @ X8 @ C2 ) ) ) ).
% UnionE
thf(fact_339_UnionE,axiom,
! [A3: nat,C2: set_set_nat] :
( ( member_nat @ A3 @ ( comple7399068483239264473et_nat @ C2 ) )
=> ~ ! [X8: set_nat] :
( ( member_nat @ A3 @ X8 )
=> ~ ( member_set_nat @ X8 @ C2 ) ) ) ).
% UnionE
thf(fact_340_UnionE,axiom,
! [A3: int,C2: set_set_int] :
( ( member_int @ A3 @ ( comple3221217463730067765et_int @ C2 ) )
=> ~ ! [X8: set_int] :
( ( member_int @ A3 @ X8 )
=> ~ ( member_set_int @ X8 @ C2 ) ) ) ).
% UnionE
thf(fact_341_UnionE,axiom,
! [A3: set_int,C2: set_set_set_int] :
( ( member_set_int @ A3 @ ( comple7281953568134767595et_int @ C2 ) )
=> ~ ! [X8: set_set_int] :
( ( member_set_int @ A3 @ X8 )
=> ~ ( member_set_set_int @ X8 @ C2 ) ) ) ).
% UnionE
thf(fact_342_int_Oint__embed__add,axiom,
! [X: int,Y: int] :
( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( plus_plus_int @ X @ Y ) )
= ( plus_plus_int @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X ) @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y ) ) ) ).
% int.int_embed_add
thf(fact_343_int_Oint__embed__one,axiom,
( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ one_one_int )
= one_one_int ) ).
% int.int_embed_one
thf(fact_344_int_Oint__embed__mult,axiom,
! [X: int,Y: int] :
( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( times_times_int @ X @ Y ) )
= ( times_times_int @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X ) @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y ) ) ) ).
% int.int_embed_mult
thf(fact_345_int_Oint__embed__zero,axiom,
( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ zero_zero_int )
= zero_zero_int ) ).
% int.int_embed_zero
thf(fact_346_int_Oint__embed__closed,axiom,
! [K2: int] : ( member_int @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K2 ) @ top_top_set_int ) ).
% int.int_embed_closed
thf(fact_347_compl__le__swap2,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ X )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_348_compl__le__swap2,axiom,
! [Y: set_set_int,X: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( uminus7346710233107665121et_int @ Y ) @ X )
=> ( ord_le4403425263959731960et_int @ ( uminus7346710233107665121et_int @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_349_compl__le__swap2,axiom,
! [Y: set_set_set_int,X: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ ( uminus3047225438891465879et_int @ Y ) @ X )
=> ( ord_le4317611570275147438et_int @ ( uminus3047225438891465879et_int @ X ) @ Y ) ) ).
% compl_le_swap2
thf(fact_350_compl__le__swap1,axiom,
! [Y: set_int,X: set_int] :
( ( ord_less_eq_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) )
=> ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).
% compl_le_swap1
thf(fact_351_compl__le__swap1,axiom,
! [Y: set_set_int,X: set_set_int] :
( ( ord_le4403425263959731960et_int @ Y @ ( uminus7346710233107665121et_int @ X ) )
=> ( ord_le4403425263959731960et_int @ X @ ( uminus7346710233107665121et_int @ Y ) ) ) ).
% compl_le_swap1
thf(fact_352_compl__le__swap1,axiom,
! [Y: set_set_set_int,X: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ Y @ ( uminus3047225438891465879et_int @ X ) )
=> ( ord_le4317611570275147438et_int @ X @ ( uminus3047225438891465879et_int @ Y ) ) ) ).
% compl_le_swap1
thf(fact_353_compl__mono,axiom,
! [X: set_int,Y: set_int] :
( ( ord_less_eq_set_int @ X @ Y )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ ( uminus1532241313380277803et_int @ X ) ) ) ).
% compl_mono
thf(fact_354_compl__mono,axiom,
! [X: set_set_int,Y: set_set_int] :
( ( ord_le4403425263959731960et_int @ X @ Y )
=> ( ord_le4403425263959731960et_int @ ( uminus7346710233107665121et_int @ Y ) @ ( uminus7346710233107665121et_int @ X ) ) ) ).
% compl_mono
thf(fact_355_compl__mono,axiom,
! [X: set_set_set_int,Y: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ X @ Y )
=> ( ord_le4317611570275147438et_int @ ( uminus3047225438891465879et_int @ Y ) @ ( uminus3047225438891465879et_int @ X ) ) ) ).
% compl_mono
thf(fact_356_int_Oint__embed__mult__aux,axiom,
! [X: int,Y: nat] :
( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( times_times_int @ X @ ( semiri1314217659103216013at_int @ Y ) ) )
= ( times_times_int @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X ) @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).
% int.int_embed_mult_aux
thf(fact_357_Sup__upper2,axiom,
! [U: int > set_int,A3: set_int_set_int,V: int > set_int] :
( ( member_int_set_int @ U @ A3 )
=> ( ( ord_le4076609313706656525et_int @ V @ U )
=> ( ord_le4076609313706656525et_int @ V @ ( comple465308567373470592et_int @ A3 ) ) ) ) ).
% Sup_upper2
thf(fact_358_Sup__upper2,axiom,
! [U: set_set_set_int,A3: set_set_set_set_int,V: set_set_set_int] :
( ( member7356822600254261989et_int @ U @ A3 )
=> ( ( ord_le4317611570275147438et_int @ V @ U )
=> ( ord_le4317611570275147438et_int @ V @ ( comple1756060948637632417et_int @ A3 ) ) ) ) ).
% Sup_upper2
thf(fact_359_Sup__upper2,axiom,
! [U: set_int,A3: set_set_int,V: set_int] :
( ( member_set_int @ U @ A3 )
=> ( ( ord_less_eq_set_int @ V @ U )
=> ( ord_less_eq_set_int @ V @ ( comple3221217463730067765et_int @ A3 ) ) ) ) ).
% Sup_upper2
thf(fact_360_Sup__upper2,axiom,
! [U: set_set_int,A3: set_set_set_int,V: set_set_int] :
( ( member_set_set_int @ U @ A3 )
=> ( ( ord_le4403425263959731960et_int @ V @ U )
=> ( ord_le4403425263959731960et_int @ V @ ( comple7281953568134767595et_int @ A3 ) ) ) ) ).
% Sup_upper2
thf(fact_361_Sup__le__iff,axiom,
! [A3: set_set_set_set_int,B: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ A3 ) @ B )
= ( ! [X5: set_set_set_int] :
( ( member7356822600254261989et_int @ X5 @ A3 )
=> ( ord_le4317611570275147438et_int @ X5 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_362_Sup__le__iff,axiom,
! [A3: set_set_int,B: set_int] :
( ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ B )
= ( ! [X5: set_int] :
( ( member_set_int @ X5 @ A3 )
=> ( ord_less_eq_set_int @ X5 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_363_Sup__le__iff,axiom,
! [A3: set_set_set_int,B: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A3 ) @ B )
= ( ! [X5: set_set_int] :
( ( member_set_set_int @ X5 @ A3 )
=> ( ord_le4403425263959731960et_int @ X5 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_364_Sup__upper,axiom,
! [X: int > set_int,A3: set_int_set_int] :
( ( member_int_set_int @ X @ A3 )
=> ( ord_le4076609313706656525et_int @ X @ ( comple465308567373470592et_int @ A3 ) ) ) ).
% Sup_upper
thf(fact_365_Sup__upper,axiom,
! [X: set_set_set_int,A3: set_set_set_set_int] :
( ( member7356822600254261989et_int @ X @ A3 )
=> ( ord_le4317611570275147438et_int @ X @ ( comple1756060948637632417et_int @ A3 ) ) ) ).
% Sup_upper
thf(fact_366_Sup__upper,axiom,
! [X: set_int,A3: set_set_int] :
( ( member_set_int @ X @ A3 )
=> ( ord_less_eq_set_int @ X @ ( comple3221217463730067765et_int @ A3 ) ) ) ).
% Sup_upper
thf(fact_367_Sup__upper,axiom,
! [X: set_set_int,A3: set_set_set_int] :
( ( member_set_set_int @ X @ A3 )
=> ( ord_le4403425263959731960et_int @ X @ ( comple7281953568134767595et_int @ A3 ) ) ) ).
% Sup_upper
thf(fact_368_Sup__least,axiom,
! [A3: set_int_set_int,Z: int > set_int] :
( ! [X4: int > set_int] :
( ( member_int_set_int @ X4 @ A3 )
=> ( ord_le4076609313706656525et_int @ X4 @ Z ) )
=> ( ord_le4076609313706656525et_int @ ( comple465308567373470592et_int @ A3 ) @ Z ) ) ).
% Sup_least
thf(fact_369_Sup__least,axiom,
! [A3: set_set_set_set_int,Z: set_set_set_int] :
( ! [X4: set_set_set_int] :
( ( member7356822600254261989et_int @ X4 @ A3 )
=> ( ord_le4317611570275147438et_int @ X4 @ Z ) )
=> ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ A3 ) @ Z ) ) ).
% Sup_least
thf(fact_370_Sup__least,axiom,
! [A3: set_set_int,Z: set_int] :
( ! [X4: set_int] :
( ( member_set_int @ X4 @ A3 )
=> ( ord_less_eq_set_int @ X4 @ Z ) )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ Z ) ) ).
% Sup_least
thf(fact_371_Sup__least,axiom,
! [A3: set_set_set_int,Z: set_set_int] :
( ! [X4: set_set_int] :
( ( member_set_set_int @ X4 @ A3 )
=> ( ord_le4403425263959731960et_int @ X4 @ Z ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A3 ) @ Z ) ) ).
% Sup_least
thf(fact_372_Sup__mono,axiom,
! [A3: set_int_set_int,B2: set_int_set_int] :
( ! [A4: int > set_int] :
( ( member_int_set_int @ A4 @ A3 )
=> ? [X2: int > set_int] :
( ( member_int_set_int @ X2 @ B2 )
& ( ord_le4076609313706656525et_int @ A4 @ X2 ) ) )
=> ( ord_le4076609313706656525et_int @ ( comple465308567373470592et_int @ A3 ) @ ( comple465308567373470592et_int @ B2 ) ) ) ).
% Sup_mono
thf(fact_373_Sup__mono,axiom,
! [A3: set_set_set_set_int,B2: set_set_set_set_int] :
( ! [A4: set_set_set_int] :
( ( member7356822600254261989et_int @ A4 @ A3 )
=> ? [X2: set_set_set_int] :
( ( member7356822600254261989et_int @ X2 @ B2 )
& ( ord_le4317611570275147438et_int @ A4 @ X2 ) ) )
=> ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ A3 ) @ ( comple1756060948637632417et_int @ B2 ) ) ) ).
% Sup_mono
thf(fact_374_Sup__mono,axiom,
! [A3: set_set_int,B2: set_set_int] :
( ! [A4: set_int] :
( ( member_set_int @ A4 @ A3 )
=> ? [X2: set_int] :
( ( member_set_int @ X2 @ B2 )
& ( ord_less_eq_set_int @ A4 @ X2 ) ) )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ ( comple3221217463730067765et_int @ B2 ) ) ) ).
% Sup_mono
thf(fact_375_Sup__mono,axiom,
! [A3: set_set_set_int,B2: set_set_set_int] :
( ! [A4: set_set_int] :
( ( member_set_set_int @ A4 @ A3 )
=> ? [X2: set_set_int] :
( ( member_set_set_int @ X2 @ B2 )
& ( ord_le4403425263959731960et_int @ A4 @ X2 ) ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A3 ) @ ( comple7281953568134767595et_int @ B2 ) ) ) ).
% Sup_mono
thf(fact_376_Sup__eqI,axiom,
! [A3: set_int_set_int,X: int > set_int] :
( ! [Y4: int > set_int] :
( ( member_int_set_int @ Y4 @ A3 )
=> ( ord_le4076609313706656525et_int @ Y4 @ X ) )
=> ( ! [Y4: int > set_int] :
( ! [Z2: int > set_int] :
( ( member_int_set_int @ Z2 @ A3 )
=> ( ord_le4076609313706656525et_int @ Z2 @ Y4 ) )
=> ( ord_le4076609313706656525et_int @ X @ Y4 ) )
=> ( ( comple465308567373470592et_int @ A3 )
= X ) ) ) ).
% Sup_eqI
thf(fact_377_Sup__eqI,axiom,
! [A3: set_set_set_set_int,X: set_set_set_int] :
( ! [Y4: set_set_set_int] :
( ( member7356822600254261989et_int @ Y4 @ A3 )
=> ( ord_le4317611570275147438et_int @ Y4 @ X ) )
=> ( ! [Y4: set_set_set_int] :
( ! [Z2: set_set_set_int] :
( ( member7356822600254261989et_int @ Z2 @ A3 )
=> ( ord_le4317611570275147438et_int @ Z2 @ Y4 ) )
=> ( ord_le4317611570275147438et_int @ X @ Y4 ) )
=> ( ( comple1756060948637632417et_int @ A3 )
= X ) ) ) ).
% Sup_eqI
thf(fact_378_Sup__eqI,axiom,
! [A3: set_set_int,X: set_int] :
( ! [Y4: set_int] :
( ( member_set_int @ Y4 @ A3 )
=> ( ord_less_eq_set_int @ Y4 @ X ) )
=> ( ! [Y4: set_int] :
( ! [Z2: set_int] :
( ( member_set_int @ Z2 @ A3 )
=> ( ord_less_eq_set_int @ Z2 @ Y4 ) )
=> ( ord_less_eq_set_int @ X @ Y4 ) )
=> ( ( comple3221217463730067765et_int @ A3 )
= X ) ) ) ).
% Sup_eqI
thf(fact_379_Sup__eqI,axiom,
! [A3: set_set_set_int,X: set_set_int] :
( ! [Y4: set_set_int] :
( ( member_set_set_int @ Y4 @ A3 )
=> ( ord_le4403425263959731960et_int @ Y4 @ X ) )
=> ( ! [Y4: set_set_int] :
( ! [Z2: set_set_int] :
( ( member_set_set_int @ Z2 @ A3 )
=> ( ord_le4403425263959731960et_int @ Z2 @ Y4 ) )
=> ( ord_le4403425263959731960et_int @ X @ Y4 ) )
=> ( ( comple7281953568134767595et_int @ A3 )
= X ) ) ) ).
% Sup_eqI
thf(fact_380_int_Oint__embed__inv,axiom,
! [X: int] :
( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( uminus_uminus_int @ X ) )
= ( uminus_uminus_int @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X ) ) ) ).
% int.int_embed_inv
thf(fact_381_Union__UNIV,axiom,
( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
= top_top_set_nat ) ).
% Union_UNIV
thf(fact_382_Union__UNIV,axiom,
( ( comple3221217463730067765et_int @ top_top_set_set_int )
= top_top_set_int ) ).
% Union_UNIV
thf(fact_383_Union__UNIV,axiom,
( ( comple7281953568134767595et_int @ top_to5524576366173240574et_int )
= top_top_set_set_int ) ).
% Union_UNIV
thf(fact_384_Union__subsetI,axiom,
! [A3: set_set_set_set_int,B2: set_set_set_set_int] :
( ! [X4: set_set_set_int] :
( ( member7356822600254261989et_int @ X4 @ A3 )
=> ? [Y5: set_set_set_int] :
( ( member7356822600254261989et_int @ Y5 @ B2 )
& ( ord_le4317611570275147438et_int @ X4 @ Y5 ) ) )
=> ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ A3 ) @ ( comple1756060948637632417et_int @ B2 ) ) ) ).
% Union_subsetI
thf(fact_385_Union__subsetI,axiom,
! [A3: set_set_int,B2: set_set_int] :
( ! [X4: set_int] :
( ( member_set_int @ X4 @ A3 )
=> ? [Y5: set_int] :
( ( member_set_int @ Y5 @ B2 )
& ( ord_less_eq_set_int @ X4 @ Y5 ) ) )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ ( comple3221217463730067765et_int @ B2 ) ) ) ).
% Union_subsetI
thf(fact_386_Union__subsetI,axiom,
! [A3: set_set_set_int,B2: set_set_set_int] :
( ! [X4: set_set_int] :
( ( member_set_set_int @ X4 @ A3 )
=> ? [Y5: set_set_int] :
( ( member_set_set_int @ Y5 @ B2 )
& ( ord_le4403425263959731960et_int @ X4 @ Y5 ) ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A3 ) @ ( comple7281953568134767595et_int @ B2 ) ) ) ).
% Union_subsetI
thf(fact_387_Union__upper,axiom,
! [B2: set_set_set_int,A3: set_set_set_set_int] :
( ( member7356822600254261989et_int @ B2 @ A3 )
=> ( ord_le4317611570275147438et_int @ B2 @ ( comple1756060948637632417et_int @ A3 ) ) ) ).
% Union_upper
thf(fact_388_Union__upper,axiom,
! [B2: set_int,A3: set_set_int] :
( ( member_set_int @ B2 @ A3 )
=> ( ord_less_eq_set_int @ B2 @ ( comple3221217463730067765et_int @ A3 ) ) ) ).
% Union_upper
thf(fact_389_Union__upper,axiom,
! [B2: set_set_int,A3: set_set_set_int] :
( ( member_set_set_int @ B2 @ A3 )
=> ( ord_le4403425263959731960et_int @ B2 @ ( comple7281953568134767595et_int @ A3 ) ) ) ).
% Union_upper
thf(fact_390_Union__least,axiom,
! [A3: set_set_set_set_int,C2: set_set_set_int] :
( ! [X8: set_set_set_int] :
( ( member7356822600254261989et_int @ X8 @ A3 )
=> ( ord_le4317611570275147438et_int @ X8 @ C2 ) )
=> ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ A3 ) @ C2 ) ) ).
% Union_least
thf(fact_391_Union__least,axiom,
! [A3: set_set_int,C2: set_int] :
( ! [X8: set_int] :
( ( member_set_int @ X8 @ A3 )
=> ( ord_less_eq_set_int @ X8 @ C2 ) )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ C2 ) ) ).
% Union_least
thf(fact_392_Union__least,axiom,
! [A3: set_set_set_int,C2: set_set_int] :
( ! [X8: set_set_int] :
( ( member_set_set_int @ X8 @ A3 )
=> ( ord_le4403425263959731960et_int @ X8 @ C2 ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A3 ) @ C2 ) ) ).
% Union_least
thf(fact_393_Union__mono,axiom,
! [A3: set_set_set_set_int,B2: set_set_set_set_int] :
( ( ord_le3130544380672780132et_int @ A3 @ B2 )
=> ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ A3 ) @ ( comple1756060948637632417et_int @ B2 ) ) ) ).
% Union_mono
thf(fact_394_Union__mono,axiom,
! [A3: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A3 @ B2 )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ ( comple3221217463730067765et_int @ B2 ) ) ) ).
% Union_mono
thf(fact_395_Union__mono,axiom,
! [A3: set_set_set_int,B2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A3 @ B2 )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A3 ) @ ( comple7281953568134767595et_int @ B2 ) ) ) ).
% Union_mono
thf(fact_396_empty__Union__conv,axiom,
! [A3: set_set_int] :
( ( bot_bot_set_int
= ( comple3221217463730067765et_int @ A3 ) )
= ( ! [X5: set_int] :
( ( member_set_int @ X5 @ A3 )
=> ( X5 = bot_bot_set_int ) ) ) ) ).
% empty_Union_conv
thf(fact_397_empty__Union__conv,axiom,
! [A3: set_set_set_int] :
( ( bot_bot_set_set_int
= ( comple7281953568134767595et_int @ A3 ) )
= ( ! [X5: set_set_int] :
( ( member_set_set_int @ X5 @ A3 )
=> ( X5 = bot_bot_set_set_int ) ) ) ) ).
% empty_Union_conv
thf(fact_398_Union__empty__conv,axiom,
! [A3: set_set_int] :
( ( ( comple3221217463730067765et_int @ A3 )
= bot_bot_set_int )
= ( ! [X5: set_int] :
( ( member_set_int @ X5 @ A3 )
=> ( X5 = bot_bot_set_int ) ) ) ) ).
% Union_empty_conv
thf(fact_399_Union__empty__conv,axiom,
! [A3: set_set_set_int] :
( ( ( comple7281953568134767595et_int @ A3 )
= bot_bot_set_set_int )
= ( ! [X5: set_set_int] :
( ( member_set_set_int @ X5 @ A3 )
=> ( X5 = bot_bot_set_set_int ) ) ) ) ).
% Union_empty_conv
thf(fact_400_Union__empty,axiom,
( ( comple3221217463730067765et_int @ bot_bot_set_set_int )
= bot_bot_set_int ) ).
% Union_empty
thf(fact_401_Union__empty,axiom,
( ( comple7281953568134767595et_int @ bot_bo2384636101374064866et_int )
= bot_bot_set_set_int ) ).
% Union_empty
thf(fact_402_int_Oring__axioms,axiom,
ring_i6310852510303722419t_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.ring_axioms
thf(fact_403_int__a__inv__eq,axiom,
! [X: int] :
( ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X )
= ( uminus_uminus_int @ X ) ) ).
% int_a_inv_eq
thf(fact_404_Sup__subset__mono,axiom,
! [A3: set_set_set_set_int,B2: set_set_set_set_int] :
( ( ord_le3130544380672780132et_int @ A3 @ B2 )
=> ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ A3 ) @ ( comple1756060948637632417et_int @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_405_Sup__subset__mono,axiom,
! [A3: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A3 @ B2 )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ ( comple3221217463730067765et_int @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_406_Sup__subset__mono,axiom,
! [A3: set_set_set_int,B2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A3 @ B2 )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A3 ) @ ( comple7281953568134767595et_int @ B2 ) ) ) ).
% Sup_subset_mono
thf(fact_407_less__eq__Sup,axiom,
! [A3: set_int_set_int,U: int > set_int] :
( ! [V2: int > set_int] :
( ( member_int_set_int @ V2 @ A3 )
=> ( ord_le4076609313706656525et_int @ U @ V2 ) )
=> ( ( A3 != bot_bo5203049024207062007et_int )
=> ( ord_le4076609313706656525et_int @ U @ ( comple465308567373470592et_int @ A3 ) ) ) ) ).
% less_eq_Sup
thf(fact_408_less__eq__Sup,axiom,
! [A3: set_set_set_set_int,U: set_set_set_int] :
( ! [V2: set_set_set_int] :
( ( member7356822600254261989et_int @ V2 @ A3 )
=> ( ord_le4317611570275147438et_int @ U @ V2 ) )
=> ( ( A3 != bot_bo2751759975269347736et_int )
=> ( ord_le4317611570275147438et_int @ U @ ( comple1756060948637632417et_int @ A3 ) ) ) ) ).
% less_eq_Sup
thf(fact_409_less__eq__Sup,axiom,
! [A3: set_set_int,U: set_int] :
( ! [V2: set_int] :
( ( member_set_int @ V2 @ A3 )
=> ( ord_less_eq_set_int @ U @ V2 ) )
=> ( ( A3 != bot_bot_set_set_int )
=> ( ord_less_eq_set_int @ U @ ( comple3221217463730067765et_int @ A3 ) ) ) ) ).
% less_eq_Sup
thf(fact_410_less__eq__Sup,axiom,
! [A3: set_set_set_int,U: set_set_int] :
( ! [V2: set_set_int] :
( ( member_set_set_int @ V2 @ A3 )
=> ( ord_le4403425263959731960et_int @ U @ V2 ) )
=> ( ( A3 != bot_bo2384636101374064866et_int )
=> ( ord_le4403425263959731960et_int @ U @ ( comple7281953568134767595et_int @ A3 ) ) ) ) ).
% less_eq_Sup
thf(fact_411_ring_Oint__embed__closed,axiom,
! [R2: partia4934656038542163276t_unit,K2: int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( member_set_int @ ( ring_i2743490682209504680t_unit @ R2 @ K2 ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ).
% ring.int_embed_closed
thf(fact_412_ring_Oint__embed__closed,axiom,
! [R2: partia2818514838349642498t_unit,K2: int] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( member_int @ ( ring_i2003929451149106142t_unit @ R2 @ K2 ) @ ( partia8426541738980984321t_unit @ R2 ) ) ) ).
% ring.int_embed_closed
thf(fact_413_ring_Oint__embed__closed,axiom,
! [R2: partia3601206958761670294t_unit,K2: int] :
( ( ring_s5321399439445376583t_unit @ R2 )
=> ( member_set_set_int @ ( ring_i2233588962022450802t_unit @ R2 @ K2 ) @ ( partia1981568256555877781t_unit @ R2 ) ) ) ).
% ring.int_embed_closed
thf(fact_414_ring_Oint__embed__closed,axiom,
! [R2: partia4692342223508353374t_unit,K2: int] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( member_nat @ ( ring_i4887507503946925882t_unit @ R2 @ K2 ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ).
% ring.int_embed_closed
thf(fact_415_ring_Oint__embed__zero,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ring_i2743490682209504680t_unit @ R2 @ zero_zero_int )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ).
% ring.int_embed_zero
thf(fact_416_ring_Oint__embed__zero,axiom,
! [R2: partia2818514838349642498t_unit] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( ( ring_i2003929451149106142t_unit @ R2 @ zero_zero_int )
= ( zero_i2266321264637750939t_unit @ R2 ) ) ) ).
% ring.int_embed_zero
thf(fact_417_ring_Oint__embed__add,axiom,
! [R2: partia4934656038542163276t_unit,X: int,Y: int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ring_i2743490682209504680t_unit @ R2 @ ( plus_plus_int @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ R2 @ ( ring_i2743490682209504680t_unit @ R2 @ X ) @ ( ring_i2743490682209504680t_unit @ R2 @ Y ) ) ) ) ).
% ring.int_embed_add
thf(fact_418_ring_Oint__embed__add,axiom,
! [R2: partia2818514838349642498t_unit,X: int,Y: int] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( ( ring_i2003929451149106142t_unit @ R2 @ ( plus_plus_int @ X @ Y ) )
= ( add_int_Product_unit @ R2 @ ( ring_i2003929451149106142t_unit @ R2 @ X ) @ ( ring_i2003929451149106142t_unit @ R2 @ Y ) ) ) ) ).
% ring.int_embed_add
thf(fact_419_s_OFactRing__iso,axiom,
( ( ( 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 ) ) ) @ i ) @ top_top_set_int )
= ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( is_rin6292643797401928902t_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 ) ) ) @ ( a_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% s.FactRing_iso
thf(fact_420_s_Othe__elem__inj,axiom,
! [X3: set_int,Y6: set_int] :
( ( member_set_int @ X3 @ ( 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 ) ) ) @ ( a_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) ) ) ) )
=> ( ( member_set_int @ Y6 @ ( 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 ) ) ) @ ( a_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) ) ) ) )
=> ( ( ( the_elem_set_int @ ( 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 ) ) ) @ i ) @ X3 ) )
= ( the_elem_set_int @ ( 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 ) ) ) @ i ) @ Y6 ) ) )
=> ( X3 = Y6 ) ) ) ) ).
% s.the_elem_inj
thf(fact_421_s_Othe__elem__wf,axiom,
! [X3: set_int] :
( ( member_set_int @ X3 @ ( 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 ) ) ) @ ( a_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) ) ) ) )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( 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 ) ) ) @ i ) @ X3 )
= ( insert_set_int @ X4 @ bot_bot_set_set_int ) ) ) ) ).
% s.the_elem_wf
thf(fact_422_s_Othe__elem__wf_H,axiom,
! [X3: set_int] :
( ( member_set_int @ X3 @ ( 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 ) ) ) @ ( a_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) ) ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ top_top_set_int )
& ( ( 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 ) ) ) @ i ) @ X3 )
= ( insert_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 ) ) ) @ i @ X4 ) @ bot_bot_set_set_int ) ) ) ) ).
% s.the_elem_wf'
thf(fact_423_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_424_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_425_i_Oa__rcos__module__minus,axiom,
! [X: int,X7: int] :
( ( ring_i6310852510303722419t_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 ) ) ) )
=> ( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ X7 @ top_top_set_int )
=> ( ( member_int @ X7 @ ( 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 ) ) ) @ i @ X ) )
= ( member_int @ ( minus_minus_int @ X7 @ X ) @ i ) ) ) ) ) ).
% i.a_rcos_module_minus
thf(fact_426_ring_Ocanonical__proj__vimage__mem__iff,axiom,
! [R2: partia3601206958761670294t_unit,I2: set_set_set_int,J: set_set_set_set_int,A: set_set_int] :
( ( ring_s5321399439445376583t_unit @ R2 )
=> ( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
=> ( ( ord_le3130544380672780132et_int @ J @ ( partia1176863786371110879t_unit @ ( factRi2699865449404500690t_unit @ R2 @ I2 ) ) )
=> ( ( member_set_set_int @ A @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( member_set_set_int @ A @ ( comple1756060948637632417et_int @ J ) )
= ( member7356822600254261989et_int @ ( a_r_co7378703583274672048t_unit @ R2 @ I2 @ A ) @ J ) ) ) ) ) ) ).
% ring.canonical_proj_vimage_mem_iff
thf(fact_427_ring_Ocanonical__proj__vimage__mem__iff,axiom,
! [R2: partia4692342223508353374t_unit,I2: set_nat,J: set_set_nat,A: nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ideal_447837613088414863t_unit @ I2 @ R2 )
=> ( ( ord_le6893508408891458716et_nat @ J @ ( partia7234930058131259047t_unit @ ( factRi8638748541043944346t_unit @ R2 @ I2 ) ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ A @ ( comple7399068483239264473et_nat @ J ) )
= ( member_set_nat @ ( a_r_co9089071853028257912t_unit @ R2 @ I2 @ A ) @ J ) ) ) ) ) ) ).
% ring.canonical_proj_vimage_mem_iff
thf(fact_428_ring_Ocanonical__proj__vimage__mem__iff,axiom,
! [R2: partia2818514838349642498t_unit,I2: set_int,J: set_set_int,A: int] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( ( ideal_6787631597145370931t_unit @ I2 @ R2 )
=> ( ( ord_le4403425263959731960et_int @ J @ ( partia966996272515721803t_unit @ ( factRi5755170488246124606t_unit @ R2 @ I2 ) ) )
=> ( ( member_int @ A @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( member_int @ A @ ( comple3221217463730067765et_int @ J ) )
= ( member_set_int @ ( a_r_co6205493800230438172t_unit @ R2 @ I2 @ A ) @ J ) ) ) ) ) ) ).
% ring.canonical_proj_vimage_mem_iff
thf(fact_429_ring_Ocanonical__proj__vimage__mem__iff,axiom,
! [R2: partia4934656038542163276t_unit,I2: set_set_int,J: set_set_set_int,A: set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
=> ( ( ord_le4317611570275147438et_int @ J @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ R2 @ I2 ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ A @ ( comple7281953568134767595et_int @ J ) )
= ( member_set_set_int @ ( a_r_co692709266861932262t_unit @ R2 @ I2 @ A ) @ J ) ) ) ) ) ) ).
% ring.canonical_proj_vimage_mem_iff
thf(fact_430_ring_Ocanonical__proj__vimage__in__carrier,axiom,
! [R2: partia3601206958761670294t_unit,I2: set_set_set_int,J: set_set_set_set_int] :
( ( ring_s5321399439445376583t_unit @ R2 )
=> ( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
=> ( ( ord_le3130544380672780132et_int @ J @ ( partia1176863786371110879t_unit @ ( factRi2699865449404500690t_unit @ R2 @ I2 ) ) )
=> ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ J ) @ ( partia1981568256555877781t_unit @ R2 ) ) ) ) ) ).
% ring.canonical_proj_vimage_in_carrier
thf(fact_431_ring_Ocanonical__proj__vimage__in__carrier,axiom,
! [R2: partia4692342223508353374t_unit,I2: set_nat,J: set_set_nat] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( ideal_447837613088414863t_unit @ I2 @ R2 )
=> ( ( ord_le6893508408891458716et_nat @ J @ ( partia7234930058131259047t_unit @ ( factRi8638748541043944346t_unit @ R2 @ I2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ J ) @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% ring.canonical_proj_vimage_in_carrier
thf(fact_432_ring_Ocanonical__proj__vimage__in__carrier,axiom,
! [R2: partia2818514838349642498t_unit,I2: set_int,J: set_set_int] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( ( ideal_6787631597145370931t_unit @ I2 @ R2 )
=> ( ( ord_le4403425263959731960et_int @ J @ ( partia966996272515721803t_unit @ ( factRi5755170488246124606t_unit @ R2 @ I2 ) ) )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ J ) @ ( partia8426541738980984321t_unit @ R2 ) ) ) ) ) ).
% ring.canonical_proj_vimage_in_carrier
thf(fact_433_ring_Ocanonical__proj__vimage__in__carrier,axiom,
! [R2: partia4934656038542163276t_unit,I2: set_set_int,J: set_set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
=> ( ( ord_le4317611570275147438et_int @ J @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ R2 @ I2 ) ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ J ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% ring.canonical_proj_vimage_in_carrier
thf(fact_434_r_Oring__ideal__imp__quot__ideal,axiom,
! [I2: set_set_int,J: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ideal_7262958097527394045t_unit @ J @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ideal_1992573625310827463t_unit @ ( image_1010086626112315521et_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 ) @ J ) @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 ) ) ) ) ).
% r.ring_ideal_imp_quot_ideal
thf(fact_435_r_Oquot__ideal__imp__ring__ideal,axiom,
! [I2: set_set_int,J: set_set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ideal_1992573625310827463t_unit @ J @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 ) )
=> ( ideal_7262958097527394045t_unit @ ( comple7281953568134767595et_int @ J ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.quot_ideal_imp_ring_ideal
thf(fact_436_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_437_r_Ocanonical__proj__vimage__mem__iff,axiom,
! [I2: set_set_int,J: set_set_set_int,A: set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ord_le4317611570275147438et_int @ J @ ( 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 @ J ) )
= ( member_set_set_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ A ) @ J ) ) ) ) ) ).
% r.canonical_proj_vimage_mem_iff
thf(fact_438_r_Ooneideal,axiom,
ideal_7262958097527394045t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).
% r.oneideal
thf(fact_439_of__nat__eq__iff,axiom,
! [M4: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M4 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M4 = N ) ) ).
% of_nat_eq_iff
thf(fact_440_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_441_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_442_mult__is__0,axiom,
! [M4: nat,N: nat] :
( ( ( times_times_nat @ M4 @ N )
= zero_zero_nat )
= ( ( M4 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_443_mult__0__right,axiom,
! [M4: nat] :
( ( times_times_nat @ M4 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_444_mult__cancel1,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M4 )
= ( times_times_nat @ K2 @ N ) )
= ( ( M4 = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_445_mult__cancel2,axiom,
! [M4: nat,K2: nat,N: nat] :
( ( ( times_times_nat @ M4 @ K2 )
= ( times_times_nat @ N @ K2 ) )
= ( ( M4 = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_446_Nat_Oadd__0__right,axiom,
! [M4: nat] :
( ( plus_plus_nat @ M4 @ zero_zero_nat )
= M4 ) ).
% Nat.add_0_right
thf(fact_447_add__is__0,axiom,
! [M4: nat,N: nat] :
( ( ( plus_plus_nat @ M4 @ N )
= zero_zero_nat )
= ( ( M4 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_448_nat__add__left__cancel__less,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M4 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M4 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_449_nat__mult__eq__1__iff,axiom,
! [M4: nat,N: nat] :
( ( ( times_times_nat @ M4 @ N )
= one_one_nat )
= ( ( M4 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_450_nat__1__eq__mult__iff,axiom,
! [M4: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M4 @ N ) )
= ( ( M4 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_451_r_Osum__zero__eq__neg,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( X
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ) ) ).
% r.sum_zero_eq_neg
thf(fact_452_r_Or__neg,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.r_neg
thf(fact_453_r_Ominus__equality,axiom,
! [Y: set_int,X: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X )
= Y ) ) ) ) ).
% r.minus_equality
thf(fact_454_r_Ol__neg,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.l_neg
thf(fact_455_r_Or__neg2,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ Y ) )
= Y ) ) ) ).
% r.r_neg2
thf(fact_456_r_Or__neg1,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= Y ) ) ) ).
% r.r_neg1
thf(fact_457_r_Ominus__add,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ) ) ).
% r.minus_add
thf(fact_458_r_Oadd_Oinv__solve__right_H,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C ) )
= A )
= ( B
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C ) ) ) ) ) ) ).
% r.add.inv_solve_right'
thf(fact_459_r_Oadd_Oinv__solve__right,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( A
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C ) ) )
= ( B
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ C ) ) ) ) ) ) ).
% r.add.inv_solve_right
thf(fact_460_r_Oadd_Oinv__solve__left_H,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B ) @ C )
= A )
= ( C
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ A ) ) ) ) ) ) ).
% r.add.inv_solve_left'
thf(fact_461_r_Oadd_Oinv__solve__left,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( A
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B ) @ C ) )
= ( C
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ B @ A ) ) ) ) ) ) ).
% r.add.inv_solve_left
thf(fact_462_r_Oadd_Oinv__mult__group,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) ) ) ) ) ).
% r.add.inv_mult_group
thf(fact_463_r_Oa__transpose__inv,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= Z )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ Z )
= Y ) ) ) ) ) ).
% r.a_transpose_inv
thf(fact_464_r_Ominus__unique,axiom,
! [Y: set_int,X: set_int,Y2: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y2 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% r.minus_unique
thf(fact_465_r_Oadd_Or__inv__ex,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ X4 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.r_inv_ex
thf(fact_466_r_Oadd_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 ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ U @ X4 )
= X4 ) )
=> ( U
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.one_unique
thf(fact_467_r_Oadd_Ol__inv__ex,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X4 @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.l_inv_ex
thf(fact_468_r_Oadd_Oinv__comm,axiom,
! [X: set_int,Y: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.add.inv_comm
thf(fact_469_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_470_r_Ocanonical__proj__vimage__in__carrier,axiom,
! [I2: set_set_int,J: set_set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ord_le4317611570275147438et_int @ J @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 ) ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ J ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.canonical_proj_vimage_in_carrier
thf(fact_471_r_Oa__r__coset__subset__G,axiom,
! [H2: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 @ X ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.a_r_coset_subset_G
thf(fact_472_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_473_r_Oint__embed__zero,axiom,
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ zero_zero_int )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.int_embed_zero
thf(fact_474_r_Oint__embed__inv,axiom,
! [X: int] :
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( uminus_uminus_int @ X ) )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) ) ) ).
% r.int_embed_inv
thf(fact_475_r_Oa__coset__add__inv2,axiom,
! [M: set_set_int,X: set_int,Y: set_int] :
( ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ X )
= ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ Y ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) )
= M ) ) ) ) ) ).
% r.a_coset_add_inv2
thf(fact_476_r_Oa__coset__add__inv1,axiom,
! [M: set_set_int,X: set_int,Y: set_int] :
( ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) )
= M )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ X )
= ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ Y ) ) ) ) ) ) ).
% r.a_coset_add_inv1
thf(fact_477_r_Oa__rcosI,axiom,
! [H: set_int,H2: set_set_int,X: set_int] :
( ( member_set_int @ H @ H2 )
=> ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H @ X ) @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 @ X ) ) ) ) ) ).
% r.a_rcosI
thf(fact_478_r_Oa__coset__add__assoc,axiom,
! [M: set_set_int,G: set_int,H: set_int] :
( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ G @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ G ) @ H )
= ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ G @ H ) ) ) ) ) ) ).
% r.a_coset_add_assoc
thf(fact_479_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_480_r_Oadd_Oone__in__subset,axiom,
! [H2: set_set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( H2 != bot_bot_set_set_int )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ H2 )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X4 ) @ H2 ) )
=> ( ! [X4: set_int] :
( ( member_set_int @ X4 @ H2 )
=> ! [Xa: set_int] :
( ( member_set_int @ Xa @ H2 )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X4 @ Xa ) @ H2 ) ) )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ H2 ) ) ) ) ) ).
% r.add.one_in_subset
thf(fact_481_of__nat__le__iff,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M4 @ N ) ) ).
% of_nat_le_iff
thf(fact_482_of__nat__le__iff,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M4 @ N ) ) ).
% of_nat_le_iff
thf(fact_483_of__nat__add,axiom,
! [M4: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M4 @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_484_of__nat__add,axiom,
! [M4: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M4 @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_485_of__nat__mult,axiom,
! [M4: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M4 @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_486_of__nat__mult,axiom,
! [M4: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M4 @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_487_nat__0__less__mult__iff,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M4 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M4 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_488_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_489_mult__less__cancel2,axiom,
! [M4: nat,K2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M4 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M4 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_490_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_491_add__gr__0,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M4 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M4 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_492_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_493_mult__le__cancel2,axiom,
! [M4: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M4 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M4 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_494_int_Ominus__closed,axiom,
! [X: int,Y: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( member_int @ Y @ top_top_set_int )
=> ( member_int @ ( minus_minus_int @ X @ Y ) @ top_top_set_int ) ) ) ).
% int.minus_closed
thf(fact_495_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_496_of__nat__eq__0__iff,axiom,
! [M4: nat] :
( ( ( semiri1316708129612266289at_nat @ M4 )
= zero_zero_nat )
= ( M4 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_497_of__nat__eq__0__iff,axiom,
! [M4: nat] :
( ( ( semiri1314217659103216013at_int @ M4 )
= zero_zero_int )
= ( M4 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_498_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_499_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_500_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_501_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_502_of__nat__less__iff,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M4 @ N ) ) ).
% of_nat_less_iff
thf(fact_503_of__nat__less__iff,axiom,
! [M4: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M4 @ N ) ) ).
% of_nat_less_iff
thf(fact_504_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_505_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_506_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_507_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_508_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_509_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_510_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_511__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Acarrier_A_IZFact_A_Iint_An_J_J_A_092_060Longrightarrow_062_Azfact__iso__inv_An_Ax_A_092_060in_062_Acarrier_A_Imod__ring_An_J_092_060close_062,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_nat @ ( zfact_iso_inv @ n @ X ) @ ( partia3499330772048238685t_unit @ ( mod_ring @ n ) ) ) ) ).
% \<open>\<And>x. x \<in> carrier (ZFact (int n)) \<Longrightarrow> zfact_iso_inv n x \<in> carrier (mod_ring n)\<close>
thf(fact_512_r_Ominus__zero,axiom,
( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.minus_zero
thf(fact_513_of__nat__le__0__iff,axiom,
! [M4: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M4 ) @ zero_zero_int )
= ( M4 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_514_of__nat__le__0__iff,axiom,
! [M4: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ zero_zero_nat )
= ( M4 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_515_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_516_r_Oadd_Oinv__eq__1__iff,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( X
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.add.inv_eq_1_iff
thf(fact_517_r_Ominus__minus,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) )
= X ) ) ).
% r.minus_minus
thf(fact_518_r_Oadd_Oinv__closed,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.add.inv_closed
thf(fact_519_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_520_s_Otrivial__hom__iff,axiom,
( ( ( 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 ) ) ) @ i ) @ top_top_set_int )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
= ( ( a_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) )
= top_top_set_int ) ) ).
% s.trivial_hom_iff
thf(fact_521_r_Or__zero,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= X ) ) ).
% r.r_zero
thf(fact_522_r_Ol__zero,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X )
= X ) ) ).
% r.l_zero
thf(fact_523_r_Oadd_Or__cancel__one_H,axiom,
! [X: set_int,A: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( X
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ X ) )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.add.r_cancel_one'
thf(fact_524_r_Oadd_Or__cancel__one,axiom,
! [X: set_int,A: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A @ X )
= X )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.add.r_cancel_one
thf(fact_525_r_Oadd_Ol__cancel__one_H,axiom,
! [X: set_int,A: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( X
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ A ) )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.add.l_cancel_one'
thf(fact_526_r_Oadd_Ol__cancel__one,axiom,
! [X: set_int,A: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ A )
= X )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ) ).
% r.add.l_cancel_one
thf(fact_527_r_Oa__coset__add__zero,axiom,
! [M: set_set_int] :
( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= M ) ) ).
% r.a_coset_add_zero
thf(fact_528_s_Okernel__zero,axiom,
! [I: int,R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit] :
( ( member_int @ I @ ( a_kern5243884610727609734t_unit @ R2 @ S @ ( 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 ) ) ) @ i ) ) )
=> ( ( 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 ) ) ) @ i @ I )
= ( zero_s6269048424454532197t_unit @ S ) ) ) ).
% s.kernel_zero
thf(fact_529_s_Ohom__zero,axiom,
( ( 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 ) ) ) @ i @ zero_zero_int )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% s.hom_zero
thf(fact_530_s_Ohom__a__inv,axiom,
! [X: int] :
( ( member_int @ X @ 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 ) ) ) @ i @ ( uminus_uminus_int @ X ) )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i @ X ) ) ) ) ).
% s.hom_a_inv
thf(fact_531_s_Othe__elem__simp,axiom,
! [X: int] :
( ( member_int @ X @ top_top_set_int )
=> ( ( the_elem_set_int @ ( 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 ) ) ) @ i ) @ ( 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_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) ) @ 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 ) ) ) @ i @ X ) ) ) ).
% s.the_elem_simp
thf(fact_532_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_533_of__nat__diff,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ N @ M4 )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M4 @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_534_of__nat__diff,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ N @ M4 )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M4 @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_535_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_536_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_537_int_Olless__eq,axiom,
( ord_less_int
= ( ^ [X5: int,Y3: int] :
( ( ord_less_eq_int @ X5 @ Y3 )
& ( X5 != Y3 ) ) ) ) ).
% int.lless_eq
thf(fact_538_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_539_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_540_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_541_ring__iso__refl,axiom,
! [R2: partia4934656038542163276t_unit] : ( is_rin6292643797401928902t_unit @ R2 @ R2 ) ).
% ring_iso_refl
thf(fact_542_ring__iso__trans,axiom,
! [R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,Q: partia4934656038542163276t_unit] :
( ( is_rin5848019822629466182t_unit @ R2 @ S )
=> ( ( is_rin1886641436590440976t_unit @ S @ Q )
=> ( is_rin1886641436590440976t_unit @ R2 @ Q ) ) ) ).
% ring_iso_trans
thf(fact_543_ring__iso__trans,axiom,
! [R2: partia4934656038542163276t_unit,S: partia2818514838349642498t_unit,Q: partia2818514838349642498t_unit] :
( ( is_rin6476721666283997948t_unit @ R2 @ S )
=> ( ( is_rin5848019822629466182t_unit @ S @ Q )
=> ( is_rin6476721666283997948t_unit @ R2 @ Q ) ) ) ).
% ring_iso_trans
thf(fact_544_ring__iso__trans,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,Q: partia2818514838349642498t_unit] :
( ( is_rin1886641436590440976t_unit @ R2 @ S )
=> ( ( is_rin6476721666283997948t_unit @ S @ Q )
=> ( is_rin5848019822629466182t_unit @ R2 @ Q ) ) ) ).
% ring_iso_trans
thf(fact_545_ring__iso__trans,axiom,
! [R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,Q: partia2818514838349642498t_unit] :
( ( is_rin6292643797401928902t_unit @ R2 @ S )
=> ( ( is_rin6476721666283997948t_unit @ S @ Q )
=> ( is_rin6476721666283997948t_unit @ R2 @ Q ) ) ) ).
% ring_iso_trans
thf(fact_546_ring__iso__trans,axiom,
! [R2: partia4934656038542163276t_unit,S: partia2818514838349642498t_unit,Q: partia4934656038542163276t_unit] :
( ( is_rin6476721666283997948t_unit @ R2 @ S )
=> ( ( is_rin1886641436590440976t_unit @ S @ Q )
=> ( is_rin6292643797401928902t_unit @ R2 @ Q ) ) ) ).
% ring_iso_trans
thf(fact_547_ring__iso__trans,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,Q: partia4934656038542163276t_unit] :
( ( is_rin1886641436590440976t_unit @ R2 @ S )
=> ( ( is_rin6292643797401928902t_unit @ S @ Q )
=> ( is_rin1886641436590440976t_unit @ R2 @ Q ) ) ) ).
% ring_iso_trans
thf(fact_548_ring__iso__trans,axiom,
! [R2: partia2818514838349642498t_unit,S: partia3601206958761670294t_unit,Q: partia4934656038542163276t_unit] :
( ( is_rin5207743797632416474t_unit @ R2 @ S )
=> ( ( is_rin1729965262454186108t_unit @ S @ Q )
=> ( is_rin1886641436590440976t_unit @ R2 @ Q ) ) ) ).
% ring_iso_trans
thf(fact_549_ring__iso__trans,axiom,
! [R2: partia3601206958761670294t_unit,S: partia2818514838349642498t_unit,Q: partia4934656038542163276t_unit] :
( ( is_rin1853299281477025458t_unit @ R2 @ S )
=> ( ( is_rin1886641436590440976t_unit @ S @ Q )
=> ( is_rin1729965262454186108t_unit @ R2 @ Q ) ) ) ).
% ring_iso_trans
thf(fact_550_ring__iso__trans,axiom,
! [R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,Q: partia4934656038542163276t_unit] :
( ( is_rin6292643797401928902t_unit @ R2 @ S )
=> ( ( is_rin6292643797401928902t_unit @ S @ Q )
=> ( is_rin6292643797401928902t_unit @ R2 @ Q ) ) ) ).
% ring_iso_trans
thf(fact_551_ring__iso__trans,axiom,
! [R2: partia3601206958761670294t_unit,S: partia4934656038542163276t_unit,Q: partia2818514838349642498t_unit] :
( ( is_rin1729965262454186108t_unit @ R2 @ S )
=> ( ( is_rin6476721666283997948t_unit @ S @ Q )
=> ( is_rin1853299281477025458t_unit @ R2 @ Q ) ) ) ).
% ring_iso_trans
thf(fact_552_add__eq__self__zero,axiom,
! [M4: nat,N: nat] :
( ( ( plus_plus_nat @ M4 @ N )
= M4 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_553_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_554_ring__iso__sym,axiom,
! [R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( is_rin6292643797401928902t_unit @ R2 @ S )
=> ( is_rin6292643797401928902t_unit @ S @ R2 ) ) ) ).
% ring_iso_sym
thf(fact_555_ring__iso__sym,axiom,
! [R2: partia3601206958761670294t_unit,S: partia4934656038542163276t_unit] :
( ( ring_s5321399439445376583t_unit @ R2 )
=> ( ( is_rin1729965262454186108t_unit @ R2 @ S )
=> ( is_rin4964436081320486288t_unit @ S @ R2 ) ) ) ).
% ring_iso_sym
thf(fact_556_ring__iso__sym,axiom,
! [R2: partia4934656038542163276t_unit,S: partia3601206958761670294t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( is_rin4964436081320486288t_unit @ R2 @ S )
=> ( is_rin1729965262454186108t_unit @ S @ R2 ) ) ) ).
% ring_iso_sym
thf(fact_557_ring__iso__sym,axiom,
! [R2: partia4934656038542163276t_unit,S: partia2818514838349642498t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( is_rin6476721666283997948t_unit @ R2 @ S )
=> ( is_rin1886641436590440976t_unit @ S @ R2 ) ) ) ).
% ring_iso_sym
thf(fact_558_ring__iso__sym,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( ( is_rin1886641436590440976t_unit @ R2 @ S )
=> ( is_rin6476721666283997948t_unit @ S @ R2 ) ) ) ).
% ring_iso_sym
thf(fact_559_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_560_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_561_ring_OFactRing__zeroideal_I1_J,axiom,
! [R2: partia2818514838349642498t_unit] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( is_rin6476721666283997948t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( insert_int @ ( zero_i2266321264637750939t_unit @ R2 ) @ bot_bot_set_int ) ) @ R2 ) ) ).
% ring.FactRing_zeroideal(1)
thf(fact_562_ring_OFactRing__zeroideal_I1_J,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( is_rin1729965262454186108t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) @ R2 ) ) ).
% ring.FactRing_zeroideal(1)
thf(fact_563_ring_OFactRing__zeroideal_I2_J,axiom,
! [R2: partia2818514838349642498t_unit] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( is_rin1886641436590440976t_unit @ R2 @ ( factRi5755170488246124606t_unit @ R2 @ ( insert_int @ ( zero_i2266321264637750939t_unit @ R2 ) @ bot_bot_set_int ) ) ) ) ).
% ring.FactRing_zeroideal(2)
thf(fact_564_ring_OFactRing__zeroideal_I2_J,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( is_rin4964436081320486288t_unit @ R2 @ ( factRi3149420076008518152t_unit @ R2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) ) ) ).
% ring.FactRing_zeroideal(2)
thf(fact_565_diff__shunt__var,axiom,
! [X: set_int,Y: set_int] :
( ( ( minus_minus_set_int @ X @ Y )
= bot_bot_set_int )
= ( ord_less_eq_set_int @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_566_diff__shunt__var,axiom,
! [X: set_set_int,Y: set_set_int] :
( ( ( minus_8897228262479074673et_int @ X @ Y )
= bot_bot_set_set_int )
= ( ord_le4403425263959731960et_int @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_567_diff__shunt__var,axiom,
! [X: set_set_set_int,Y: set_set_set_int] :
( ( ( minus_6857623457997529383et_int @ X @ Y )
= bot_bo2384636101374064866et_int )
= ( ord_le4317611570275147438et_int @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_568_int_Ominus__eq,axiom,
( minus_minus_int
= ( ^ [X5: int,Y3: int] : ( plus_plus_int @ X5 @ ( uminus_uminus_int @ Y3 ) ) ) ) ).
% int.minus_eq
thf(fact_569_nat__neq__iff,axiom,
! [M4: nat,N: nat] :
( ( M4 != N )
= ( ( ord_less_nat @ M4 @ N )
| ( ord_less_nat @ N @ M4 ) ) ) ).
% nat_neq_iff
thf(fact_570_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_571_less__not__refl2,axiom,
! [N: nat,M4: nat] :
( ( ord_less_nat @ N @ M4 )
=> ( M4 != N ) ) ).
% less_not_refl2
thf(fact_572_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_573_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_574_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_575_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_576_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_577_less__imp__add__positive,axiom,
! [I: nat,J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J3 ) ) ) ).
% less_imp_add_positive
thf(fact_578_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_579_less__add__eq__less,axiom,
! [K2: nat,L: nat,M4: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M4 @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M4 @ N ) ) ) ).
% less_add_eq_less
thf(fact_580_trans__less__add2,axiom,
! [I: nat,J3: nat,M4: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M4 @ J3 ) ) ) ).
% trans_less_add2
thf(fact_581_trans__less__add1,axiom,
! [I: nat,J3: nat,M4: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J3 @ M4 ) ) ) ).
% trans_less_add1
thf(fact_582_mult__less__mono2,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J3 ) ) ) ) ).
% mult_less_mono2
thf(fact_583_mult__less__mono1,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J3 @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_584_gr__implies__not0,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_585_add__less__mono1,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J3 @ K2 ) ) ) ).
% add_less_mono1
thf(fact_586_not__add__less2,axiom,
! [J3: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J3 @ I ) @ I ) ).
% not_add_less2
thf(fact_587_not__add__less1,axiom,
! [I: nat,J3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J3 ) @ I ) ).
% not_add_less1
thf(fact_588_add__less__mono,axiom,
! [I: nat,J3: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_less_mono
thf(fact_589_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_590_add__lessD1,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J3 ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_591_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_592_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_593_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_594_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_595_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N3: nat] :
( ( ord_less_eq_nat @ M6 @ N3 )
& ( M6 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_596_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_597_less__imp__le__nat,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% less_imp_le_nat
thf(fact_598_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
| ( M6 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_599_less__or__eq__imp__le,axiom,
! [M4: nat,N: nat] :
( ( ( ord_less_nat @ M4 @ N )
| ( M4 = N ) )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_600_mono__nat__linear__lb,axiom,
! [F: nat > nat,M4: nat,K2: nat] :
( ! [M7: nat,N2: nat] :
( ( ord_less_nat @ M7 @ N2 )
=> ( ord_less_nat @ ( F @ M7 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M4 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_601_le__neq__implies__less,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( M4 != N )
=> ( ord_less_nat @ M4 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_602_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J3: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_603_mult__eq__self__implies__10,axiom,
! [M4: nat,N: nat] :
( ( M4
= ( times_times_nat @ M4 @ N ) )
=> ( ( N = one_one_nat )
| ( M4 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_604_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_605_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_606_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_607_ring__hom__ring_Otrivial__hom__iff,axiom,
! [R2: partia4934656038542163276t_unit,S: partia2818514838349642498t_unit,H: set_int > int] :
( ( ring_h8685743689922412466t_unit @ R2 @ S @ H )
=> ( ( ( image_set_int_int @ H @ ( partia966996272515721803t_unit @ R2 ) )
= ( insert_int @ ( zero_i2266321264637750939t_unit @ S ) @ bot_bot_set_int ) )
= ( ( a_kern610592803566390898t_unit @ R2 @ S @ H )
= ( partia966996272515721803t_unit @ R2 ) ) ) ) ).
% ring_hom_ring.trivial_hom_iff
thf(fact_608_ring__hom__ring_Otrivial__hom__iff,axiom,
! [R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,H: set_int > set_int] :
( ( ring_h98094481560215932t_unit @ R2 @ S @ H )
=> ( ( ( image_524474410958335435et_int @ H @ ( partia966996272515721803t_unit @ R2 ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ S ) @ bot_bot_set_set_int ) )
= ( ( a_kern4456122948206014524t_unit @ R2 @ S @ H )
= ( partia966996272515721803t_unit @ R2 ) ) ) ) ).
% ring_hom_ring.trivial_hom_iff
thf(fact_609_ring__hom__ring_Otrivial__hom__iff,axiom,
! [R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,H: int > int] :
( ( ring_h1606527851073305852t_unit @ R2 @ S @ H )
=> ( ( ( image_int_int @ H @ ( partia8426541738980984321t_unit @ R2 ) )
= ( insert_int @ ( zero_i2266321264637750939t_unit @ S ) @ bot_bot_set_int ) )
= ( ( a_kern6030145918964829116t_unit @ R2 @ S @ H )
= ( partia8426541738980984321t_unit @ R2 ) ) ) ) ).
% ring_hom_ring.trivial_hom_iff
thf(fact_610_ring__hom__ring_Otrivial__hom__iff,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,H: int > set_int] :
( ( ring_h4095663460228855494t_unit @ R2 @ S @ H )
=> ( ( ( image_int_set_int @ H @ ( partia8426541738980984321t_unit @ R2 ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ S ) @ bot_bot_set_set_int ) )
= ( ( a_kern5243884610727609734t_unit @ R2 @ S @ H )
= ( partia8426541738980984321t_unit @ R2 ) ) ) ) ).
% ring_hom_ring.trivial_hom_iff
thf(fact_611_ring__hom__ring_Otrivial__hom__iff,axiom,
! [R2: partia3601206958761670294t_unit,S: partia2818514838349642498t_unit,H: set_set_int > int] :
( ( ring_h4882122002490088296t_unit @ R2 @ S @ H )
=> ( ( ( image_4671129442152397515nt_int @ H @ ( partia1981568256555877781t_unit @ R2 ) )
= ( insert_int @ ( zero_i2266321264637750939t_unit @ S ) @ bot_bot_set_int ) )
= ( ( a_kern16778432281111080t_unit @ R2 @ S @ H )
= ( partia1981568256555877781t_unit @ R2 ) ) ) ) ).
% ring_hom_ring.trivial_hom_iff
thf(fact_612_ring__hom__ring_Otrivial__hom__iff,axiom,
! [R2: partia3601206958761670294t_unit,S: partia4934656038542163276t_unit,H: set_set_int > set_int] :
( ( ring_h8546004371043296562t_unit @ R2 @ S @ H )
=> ( ( ( image_3513010637850279041et_int @ H @ ( partia1981568256555877781t_unit @ R2 ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ S ) @ bot_bot_set_set_int ) )
= ( ( a_kern4580734159119725554t_unit @ R2 @ S @ H )
= ( partia1981568256555877781t_unit @ R2 ) ) ) ) ).
% ring_hom_ring.trivial_hom_iff
thf(fact_613_ring__hom__ring_Otrivial__hom__iff,axiom,
! [R2: partia4692342223508353374t_unit,S: partia2818514838349642498t_unit,H: nat > int] :
( ( ring_h3200931406939544736t_unit @ R2 @ S @ H )
=> ( ( ( image_nat_int @ H @ ( partia3499330772048238685t_unit @ R2 ) )
= ( insert_int @ ( zero_i2266321264637750939t_unit @ S ) @ bot_bot_set_int ) )
= ( ( a_kern7624549474831068000t_unit @ R2 @ S @ H )
= ( partia3499330772048238685t_unit @ R2 ) ) ) ) ).
% ring_hom_ring.trivial_hom_iff
thf(fact_614_ring__hom__ring_Otrivial__hom__iff,axiom,
! [R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,H: nat > set_int] :
( ( ring_h7271012355634579050t_unit @ R2 @ S @ H )
=> ( ( ( image_nat_set_int @ H @ ( partia3499330772048238685t_unit @ R2 ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ S ) @ bot_bot_set_set_int ) )
= ( ( a_kern8419233506133333290t_unit @ R2 @ S @ H )
= ( partia3499330772048238685t_unit @ R2 ) ) ) ) ).
% ring_hom_ring.trivial_hom_iff
thf(fact_615_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_616_ring__hom__ring_Othe__elem__simp,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,H: int > set_int,X: int] :
( ( ring_h4095663460228855494t_unit @ R2 @ S @ H )
=> ( ( member_int @ X @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( the_elem_set_int @ ( image_int_set_int @ H @ ( a_r_co6205493800230438172t_unit @ R2 @ ( a_kern5243884610727609734t_unit @ R2 @ S @ H ) @ X ) ) )
= ( H @ X ) ) ) ) ).
% ring_hom_ring.the_elem_simp
thf(fact_617_ring__hom__ring_Othe__elem__simp,axiom,
! [R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,H: int > int,X: int] :
( ( ring_h1606527851073305852t_unit @ R2 @ S @ H )
=> ( ( member_int @ X @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( the_elem_int @ ( image_int_int @ H @ ( a_r_co6205493800230438172t_unit @ R2 @ ( a_kern6030145918964829116t_unit @ R2 @ S @ H ) @ X ) ) )
= ( H @ X ) ) ) ) ).
% ring_hom_ring.the_elem_simp
thf(fact_618_ring__hom__ring_Othe__elem__inj,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,H: int > set_int,X3: set_int,Y6: set_int] :
( ( ring_h4095663460228855494t_unit @ R2 @ S @ H )
=> ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( a_kern5243884610727609734t_unit @ R2 @ S @ H ) ) ) )
=> ( ( member_set_int @ Y6 @ ( partia966996272515721803t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( a_kern5243884610727609734t_unit @ R2 @ S @ H ) ) ) )
=> ( ( ( the_elem_set_int @ ( image_int_set_int @ H @ X3 ) )
= ( the_elem_set_int @ ( image_int_set_int @ H @ Y6 ) ) )
=> ( X3 = Y6 ) ) ) ) ) ).
% ring_hom_ring.the_elem_inj
thf(fact_619_ring__hom__ring_Othe__elem__inj,axiom,
! [R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,H: int > int,X3: set_int,Y6: set_int] :
( ( ring_h1606527851073305852t_unit @ R2 @ S @ H )
=> ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( a_kern6030145918964829116t_unit @ R2 @ S @ H ) ) ) )
=> ( ( member_set_int @ Y6 @ ( partia966996272515721803t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( a_kern6030145918964829116t_unit @ R2 @ S @ H ) ) ) )
=> ( ( ( the_elem_int @ ( image_int_int @ H @ X3 ) )
= ( the_elem_int @ ( image_int_int @ H @ Y6 ) ) )
=> ( X3 = Y6 ) ) ) ) ) ).
% ring_hom_ring.the_elem_inj
thf(fact_620_ring__hom__ring_Othe__elem__wf,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,H: int > set_int,X3: set_int] :
( ( ring_h4095663460228855494t_unit @ R2 @ S @ H )
=> ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( a_kern5243884610727609734t_unit @ R2 @ S @ H ) ) ) )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ S ) )
& ( ( image_int_set_int @ H @ X3 )
= ( insert_set_int @ X4 @ bot_bot_set_set_int ) ) ) ) ) ).
% ring_hom_ring.the_elem_wf
thf(fact_621_ring__hom__ring_Othe__elem__wf,axiom,
! [R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,H: int > int,X3: set_int] :
( ( ring_h1606527851073305852t_unit @ R2 @ S @ H )
=> ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( a_kern6030145918964829116t_unit @ R2 @ S @ H ) ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ ( partia8426541738980984321t_unit @ S ) )
& ( ( image_int_int @ H @ X3 )
= ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ) ).
% ring_hom_ring.the_elem_wf
thf(fact_622_ring__hom__ring_Othe__elem__wf,axiom,
! [R2: partia2818514838349642498t_unit,S: partia3601206958761670294t_unit,H: int > set_set_int,X3: set_int] :
( ( ring_h8236566518645479312t_unit @ R2 @ S @ H )
=> ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( a_kern3371222948436502096t_unit @ R2 @ S @ H ) ) ) )
=> ? [X4: set_set_int] :
( ( member_set_set_int @ X4 @ ( partia1981568256555877781t_unit @ S ) )
& ( ( image_6617583118289273547et_int @ H @ X3 )
= ( insert_set_set_int @ X4 @ bot_bo2384636101374064866et_int ) ) ) ) ) ).
% ring_hom_ring.the_elem_wf
thf(fact_623_ring__hom__ring_Othe__elem__wf,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4692342223508353374t_unit,H: int > nat,X3: set_int] :
( ( ring_h4490105903871125592t_unit @ R2 @ S @ H )
=> ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( a_kern8913723971762648856t_unit @ R2 @ S @ H ) ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ S ) )
& ( ( image_int_nat @ H @ X3 )
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% ring_hom_ring.the_elem_wf
thf(fact_624_ring__hom__ring_Othe__elem__wf,axiom,
! [R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,H: set_int > set_int,X3: set_set_int] :
( ( ring_h98094481560215932t_unit @ R2 @ S @ H )
=> ( ( member_set_set_int @ X3 @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern4456122948206014524t_unit @ R2 @ S @ H ) ) ) )
=> ? [X4: set_int] :
( ( member_set_int @ X4 @ ( partia966996272515721803t_unit @ S ) )
& ( ( image_524474410958335435et_int @ H @ X3 )
= ( insert_set_int @ X4 @ bot_bot_set_set_int ) ) ) ) ) ).
% ring_hom_ring.the_elem_wf
thf(fact_625_ring__hom__ring_Othe__elem__wf,axiom,
! [R2: partia4934656038542163276t_unit,S: partia2818514838349642498t_unit,H: set_int > int,X3: set_set_int] :
( ( ring_h8685743689922412466t_unit @ R2 @ S @ H )
=> ( ( member_set_set_int @ X3 @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern610592803566390898t_unit @ R2 @ S @ H ) ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ ( partia8426541738980984321t_unit @ S ) )
& ( ( image_set_int_int @ H @ X3 )
= ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ) ).
% ring_hom_ring.the_elem_wf
thf(fact_626_ring__hom__ring_Othe__elem__wf,axiom,
! [R2: partia4934656038542163276t_unit,S: partia3601206958761670294t_unit,H: set_int > set_set_int,X3: set_set_int] :
( ( ring_h2557103153054820934t_unit @ R2 @ S @ H )
=> ( ( member_set_set_int @ X3 @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern7815204977986025734t_unit @ R2 @ S @ H ) ) ) )
=> ? [X4: set_set_int] :
( ( member_set_set_int @ X4 @ ( partia1981568256555877781t_unit @ S ) )
& ( ( image_1010086626112315521et_int @ H @ X3 )
= ( insert_set_set_int @ X4 @ bot_bo2384636101374064866et_int ) ) ) ) ) ).
% ring_hom_ring.the_elem_wf
thf(fact_627_ring__hom__ring_Othe__elem__wf,axiom,
! [R2: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,H: set_int > nat,X3: set_set_int] :
( ( ring_h2345949705865456398t_unit @ R2 @ S @ H )
=> ( ( member_set_set_int @ X3 @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern3494170856364210638t_unit @ R2 @ S @ H ) ) ) )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( partia3499330772048238685t_unit @ S ) )
& ( ( image_set_int_nat @ H @ X3 )
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% ring_hom_ring.the_elem_wf
thf(fact_628_ring__hom__ring_Othe__elem__wf_H,axiom,
! [R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,H: int > int,X3: set_int] :
( ( ring_h1606527851073305852t_unit @ R2 @ S @ H )
=> ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( a_kern6030145918964829116t_unit @ R2 @ S @ H ) ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ ( partia8426541738980984321t_unit @ R2 ) )
& ( ( image_int_int @ H @ X3 )
= ( insert_int @ ( H @ X4 ) @ bot_bot_set_int ) ) ) ) ) ).
% ring_hom_ring.the_elem_wf'
thf(fact_629_ring__hom__ring_Othe__elem__wf_H,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,H: int > set_int,X3: set_int] :
( ( ring_h4095663460228855494t_unit @ R2 @ S @ H )
=> ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( a_kern5243884610727609734t_unit @ R2 @ S @ H ) ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ ( partia8426541738980984321t_unit @ R2 ) )
& ( ( image_int_set_int @ H @ X3 )
= ( insert_set_int @ ( H @ X4 ) @ bot_bot_set_set_int ) ) ) ) ) ).
% ring_hom_ring.the_elem_wf'
thf(fact_630_ring__hom__ring_OFactRing__iso,axiom,
! [R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,H: int > int] :
( ( ring_h1606527851073305852t_unit @ R2 @ S @ H )
=> ( ( ( image_int_int @ H @ ( partia8426541738980984321t_unit @ R2 ) )
= ( partia8426541738980984321t_unit @ S ) )
=> ( is_rin6476721666283997948t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( a_kern6030145918964829116t_unit @ R2 @ S @ H ) ) @ S ) ) ) ).
% ring_hom_ring.FactRing_iso
thf(fact_631_ring__hom__ring_OFactRing__iso,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4692342223508353374t_unit,H: int > nat] :
( ( ring_h4490105903871125592t_unit @ R2 @ S @ H )
=> ( ( ( image_int_nat @ H @ ( partia8426541738980984321t_unit @ R2 ) )
= ( partia3499330772048238685t_unit @ S ) )
=> ( is_rin136927682227041880t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( a_kern8913723971762648856t_unit @ R2 @ S @ H ) ) @ S ) ) ) ).
% ring_hom_ring.FactRing_iso
thf(fact_632_ring__hom__ring_OFactRing__iso,axiom,
! [R2: partia4692342223508353374t_unit,S: partia2818514838349642498t_unit,H: nat > int] :
( ( ring_h3200931406939544736t_unit @ R2 @ S @ H )
=> ( ( ( image_nat_int @ H @ ( partia3499330772048238685t_unit @ R2 ) )
= ( partia8426541738980984321t_unit @ S ) )
=> ( is_rin8002773474025691296t_unit @ ( factRi8638748541043944346t_unit @ R2 @ ( a_kern7624549474831068000t_unit @ R2 @ S @ H ) ) @ S ) ) ) ).
% ring_hom_ring.FactRing_iso
thf(fact_633_ring__hom__ring_OFactRing__iso,axiom,
! [R2: partia4692342223508353374t_unit,S: partia4692342223508353374t_unit,H: nat > nat] :
( ( ring_h6084509459737364476t_unit @ R2 @ S @ H )
=> ( ( ( image_nat_nat @ H @ ( partia3499330772048238685t_unit @ R2 ) )
= ( partia3499330772048238685t_unit @ S ) )
=> ( is_rin1662979489968735228t_unit @ ( factRi8638748541043944346t_unit @ R2 @ ( a_kern1284755490774111932t_unit @ R2 @ S @ H ) ) @ S ) ) ) ).
% ring_hom_ring.FactRing_iso
thf(fact_634_ring__hom__ring_OFactRing__iso,axiom,
! [R2: partia4934656038542163276t_unit,S: partia2818514838349642498t_unit,H: set_int > int] :
( ( ring_h8685743689922412466t_unit @ R2 @ S @ H )
=> ( ( ( image_set_int_int @ H @ ( partia966996272515721803t_unit @ R2 ) )
= ( partia8426541738980984321t_unit @ S ) )
=> ( is_rin1853299281477025458t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern610592803566390898t_unit @ R2 @ S @ H ) ) @ S ) ) ) ).
% ring_hom_ring.FactRing_iso
thf(fact_635_ring__hom__ring_OFactRing__iso,axiom,
! [R2: partia4934656038542163276t_unit,S: partia4692342223508353374t_unit,H: set_int > nat] :
( ( ring_h2345949705865456398t_unit @ R2 @ S @ H )
=> ( ( ( image_set_int_nat @ H @ ( partia966996272515721803t_unit @ R2 ) )
= ( partia3499330772048238685t_unit @ S ) )
=> ( is_rin4736877334274845198t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern3494170856364210638t_unit @ R2 @ S @ H ) ) @ S ) ) ) ).
% ring_hom_ring.FactRing_iso
thf(fact_636_ring__hom__ring_OFactRing__iso,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,H: int > set_int] :
( ( ring_h4095663460228855494t_unit @ R2 @ S @ H )
=> ( ( ( image_int_set_int @ H @ ( partia8426541738980984321t_unit @ R2 ) )
= ( partia966996272515721803t_unit @ S ) )
=> ( is_rin6292643797401928902t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( a_kern5243884610727609734t_unit @ R2 @ S @ H ) ) @ S ) ) ) ).
% ring_hom_ring.FactRing_iso
thf(fact_637_ring__hom__ring_OFactRing__iso,axiom,
! [R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,H: nat > set_int] :
( ( ring_h7271012355634579050t_unit @ R2 @ S @ H )
=> ( ( ( image_nat_set_int @ H @ ( partia3499330772048238685t_unit @ R2 ) )
= ( partia966996272515721803t_unit @ S ) )
=> ( is_rin8841082319626682986t_unit @ ( factRi8638748541043944346t_unit @ R2 @ ( a_kern8419233506133333290t_unit @ R2 @ S @ H ) ) @ S ) ) ) ).
% ring_hom_ring.FactRing_iso
thf(fact_638_ring__hom__ring_OFactRing__iso,axiom,
! [R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,H: set_int > set_int] :
( ( ring_h98094481560215932t_unit @ R2 @ S @ H )
=> ( ( ( image_524474410958335435et_int @ H @ ( partia966996272515721803t_unit @ R2 ) )
= ( partia966996272515721803t_unit @ S ) )
=> ( is_rin1729965262454186108t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern4456122948206014524t_unit @ R2 @ S @ H ) ) @ S ) ) ) ).
% ring_hom_ring.FactRing_iso
thf(fact_639_ring__hom__ring_OFactRing__iso,axiom,
! [R2: partia2818514838349642498t_unit,S: partia3601206958761670294t_unit,H: int > set_set_int] :
( ( ring_h8236566518645479312t_unit @ R2 @ S @ H )
=> ( ( ( image_6617583118289273547et_int @ H @ ( partia8426541738980984321t_unit @ R2 ) )
= ( partia1981568256555877781t_unit @ S ) )
=> ( is_rin4964436081320486288t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( a_kern3371222948436502096t_unit @ R2 @ S @ H ) ) @ S ) ) ) ).
% ring_hom_ring.FactRing_iso
thf(fact_640_of__nat__mono,axiom,
! [I: nat,J3: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J3 ) ) ) ).
% of_nat_mono
thf(fact_641_of__nat__mono,axiom,
! [I: nat,J3: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J3 ) ) ) ).
% of_nat_mono
thf(fact_642_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_643_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_644_int_Oint__embed__diff,axiom,
! [X: int,Y: int] :
( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_int @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X ) @ ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y ) ) ) ).
% int.int_embed_diff
thf(fact_645_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_646_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_647_of__nat__less__0__iff,axiom,
! [M4: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_648_of__nat__less__0__iff,axiom,
! [M4: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M4 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_649_less__imp__of__nat__less,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_650_less__imp__of__nat__less,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_651_of__nat__less__imp__less,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M4 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M4 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_652_of__nat__less__imp__less,axiom,
! [M4: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M4 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_653_int_Oquotient__eq__iff__same__a__r__cos,axiom,
! [I2: set_int,A: int,B: 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 @ A @ top_top_set_int )
=> ( ( member_int @ B @ top_top_set_int )
=> ( ( member_int @ ( minus_minus_int @ A @ B ) @ 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 @ A )
= ( 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 @ B ) ) ) ) ) ) ).
% int.quotient_eq_iff_same_a_r_cos
thf(fact_654_ring_Oring__hom__restrict,axiom,
! [R2: partia2818514838349642498t_unit,F: int > set_int,S: partia4934656038542163276t_unit,G: int > set_int] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( ( member_int_set_int @ F @ ( ring_h1577560673974712708t_unit @ R2 @ S ) )
=> ( ! [R: int] :
( ( member_int @ R @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( F @ R )
= ( G @ R ) ) )
=> ( member_int_set_int @ G @ ( ring_h1577560673974712708t_unit @ R2 @ S ) ) ) ) ) ).
% ring.ring_hom_restrict
thf(fact_655_int_Oquot__ideal__imp__ring__ideal,axiom,
! [I2: set_int,J: 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 @ J @ ( 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 @ J ) @ ( 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_656_int_Oring__ideal__imp__quot__ideal,axiom,
! [I2: set_int,J: 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 @ J @ ( 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 ) @ J ) @ ( 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_657_ideal_Oquotient__is__ring,axiom,
! [I2: set_set_set_int,R2: partia3601206958761670294t_unit] :
( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
=> ( ring_s3163219825908230673t_unit @ ( factRi2699865449404500690t_unit @ R2 @ I2 ) ) ) ).
% ideal.quotient_is_ring
thf(fact_658_ideal_Oquotient__is__ring,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( ideal_6787631597145370931t_unit @ I2 @ R2 )
=> ( ring_s5316885176909347197t_unit @ ( factRi5755170488246124606t_unit @ R2 @ I2 ) ) ) ).
% ideal.quotient_is_ring
thf(fact_659_ideal_Oquotient__is__ring,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
=> ( ring_s5321399439445376583t_unit @ ( factRi3149420076008518152t_unit @ R2 @ I2 ) ) ) ).
% ideal.quotient_is_ring
thf(fact_660_ring_Oquot__ideal__imp__ring__ideal,axiom,
! [R2: partia3601206958761670294t_unit,I2: set_set_set_int,J: set_set_set_set_int] :
( ( ring_s5321399439445376583t_unit @ R2 )
=> ( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
=> ( ( ideal_6096881031695155089t_unit @ J @ ( factRi2699865449404500690t_unit @ R2 @ I2 ) )
=> ( ideal_1992573625310827463t_unit @ ( comple1756060948637632417et_int @ J ) @ R2 ) ) ) ) ).
% ring.quot_ideal_imp_ring_ideal
thf(fact_661_ring_Oquot__ideal__imp__ring__ideal,axiom,
! [R2: partia2818514838349642498t_unit,I2: set_int,J: set_set_int] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( ( ideal_6787631597145370931t_unit @ I2 @ R2 )
=> ( ( ideal_7262958097527394045t_unit @ J @ ( factRi5755170488246124606t_unit @ R2 @ I2 ) )
=> ( ideal_6787631597145370931t_unit @ ( comple3221217463730067765et_int @ J ) @ R2 ) ) ) ) ).
% ring.quot_ideal_imp_ring_ideal
thf(fact_662_ring_Oquot__ideal__imp__ring__ideal,axiom,
! [R2: partia4934656038542163276t_unit,I2: set_set_int,J: set_set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
=> ( ( ideal_1992573625310827463t_unit @ J @ ( factRi3149420076008518152t_unit @ R2 @ I2 ) )
=> ( ideal_7262958097527394045t_unit @ ( comple7281953568134767595et_int @ J ) @ R2 ) ) ) ) ).
% ring.quot_ideal_imp_ring_ideal
thf(fact_663_ideal_Orcos__ring__hom,axiom,
! [I2: set_set_set_int,R2: partia3601206958761670294t_unit] :
( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
=> ( member5110176937812460956et_int @ ( a_r_co7378703583274672048t_unit @ R2 @ I2 ) @ ( ring_h7502273849718182020t_unit @ R2 @ ( factRi2699865449404500690t_unit @ R2 @ I2 ) ) ) ) ).
% ideal.rcos_ring_hom
thf(fact_664_ideal_Orcos__ring__hom,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( ideal_6787631597145370931t_unit @ I2 @ R2 )
=> ( member_int_set_int @ ( a_r_co6205493800230438172t_unit @ R2 @ I2 ) @ ( ring_h1577560673974712708t_unit @ R2 @ ( factRi5755170488246124606t_unit @ R2 @ I2 ) ) ) ) ).
% ideal.rcos_ring_hom
thf(fact_665_ideal_Orcos__ring__hom,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
=> ( member1123395007370249776et_int @ ( a_r_co692709266861932262t_unit @ R2 @ I2 ) @ ( ring_h2644251554410904836t_unit @ R2 @ ( factRi3149420076008518152t_unit @ R2 @ I2 ) ) ) ) ).
% ideal.rcos_ring_hom
thf(fact_666_ideal_Orcos__ring__hom__ring,axiom,
! [I2: set_set_set_int,R2: partia3601206958761670294t_unit] :
( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
=> ( ring_h8406274701427011014t_unit @ R2 @ ( factRi2699865449404500690t_unit @ R2 @ I2 ) @ ( a_r_co7378703583274672048t_unit @ R2 @ I2 ) ) ) ).
% ideal.rcos_ring_hom_ring
thf(fact_667_ideal_Orcos__ring__hom__ring,axiom,
! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
=> ( ring_h2557103153054820934t_unit @ R2 @ ( factRi3149420076008518152t_unit @ R2 @ I2 ) @ ( a_r_co692709266861932262t_unit @ R2 @ I2 ) ) ) ).
% ideal.rcos_ring_hom_ring
thf(fact_668_ideal_Orcos__ring__hom__ring,axiom,
! [I2: set_int,R2: partia2818514838349642498t_unit] :
( ( ideal_6787631597145370931t_unit @ I2 @ R2 )
=> ( ring_h4095663460228855494t_unit @ R2 @ ( factRi5755170488246124606t_unit @ R2 @ I2 ) @ ( a_r_co6205493800230438172t_unit @ R2 @ I2 ) ) ) ).
% ideal.rcos_ring_hom_ring
thf(fact_669_a__coset__hom_I2_J,axiom,
! [H: set_int > int,R2: partia4934656038542163276t_unit,S: partia2818514838349642498t_unit,I2: set_set_int,A: set_int] :
( ( member_set_int_int @ H @ ( ring_h6167640903668269680t_unit @ R2 @ S ) )
=> ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( image_set_int_int @ H @ ( a_r_co692709266861932262t_unit @ R2 @ I2 @ A ) )
= ( a_r_co6205493800230438172t_unit @ S @ ( image_set_int_int @ H @ I2 ) @ ( H @ A ) ) ) ) ) ) ).
% a_coset_hom(2)
thf(fact_670_a__coset__hom_I2_J,axiom,
! [H: set_int > set_int,R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,I2: set_set_int,A: set_int] :
( ( member5205197933313416826et_int @ H @ ( ring_h3404898052528352314t_unit @ R2 @ S ) )
=> ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( image_524474410958335435et_int @ H @ ( a_r_co692709266861932262t_unit @ R2 @ I2 @ A ) )
= ( a_r_co692709266861932262t_unit @ S @ ( image_524474410958335435et_int @ H @ I2 ) @ ( H @ A ) ) ) ) ) ) ).
% a_coset_hom(2)
thf(fact_671_a__coset__hom_I2_J,axiom,
! [H: int > int,R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,I2: set_int,A: int] :
( ( member_int_int @ H @ ( ring_h9157551604493154234t_unit @ R2 @ S ) )
=> ( ( ord_less_eq_set_int @ I2 @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( member_int @ A @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( image_int_int @ H @ ( a_r_co6205493800230438172t_unit @ R2 @ I2 @ A ) )
= ( a_r_co6205493800230438172t_unit @ S @ ( image_int_int @ H @ I2 ) @ ( H @ A ) ) ) ) ) ) ).
% a_coset_hom(2)
thf(fact_672_a__coset__hom_I2_J,axiom,
! [H: int > set_int,R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,I2: set_int,A: int] :
( ( member_int_set_int @ H @ ( ring_h1577560673974712708t_unit @ R2 @ S ) )
=> ( ( ord_less_eq_set_int @ I2 @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( member_int @ A @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( image_int_set_int @ H @ ( a_r_co6205493800230438172t_unit @ R2 @ I2 @ A ) )
= ( a_r_co692709266861932262t_unit @ S @ ( image_int_set_int @ H @ I2 ) @ ( H @ A ) ) ) ) ) ) ).
% a_coset_hom(2)
thf(fact_673_a__coset__hom_I2_J,axiom,
! [H: set_set_int > int,R2: partia3601206958761670294t_unit,S: partia2818514838349642498t_unit,I2: set_set_set_int,A: set_set_int] :
( ( member2922251196090340986nt_int @ H @ ( ring_h8188925573458224678t_unit @ R2 @ S ) )
=> ( ( ord_le4317611570275147438et_int @ I2 @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( member_set_set_int @ A @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( image_4671129442152397515nt_int @ H @ ( a_r_co7378703583274672048t_unit @ R2 @ I2 @ A ) )
= ( a_r_co6205493800230438172t_unit @ S @ ( image_4671129442152397515nt_int @ H @ I2 ) @ ( H @ A ) ) ) ) ) ) ).
% a_coset_hom(2)
thf(fact_674_a__coset__hom_I2_J,axiom,
! [H: set_set_int > set_int,R2: partia3601206958761670294t_unit,S: partia4934656038542163276t_unit,I2: set_set_set_int,A: set_set_int] :
( ( member2198268278526348592et_int @ H @ ( ring_h8633152772399380464t_unit @ R2 @ S ) )
=> ( ( ord_le4317611570275147438et_int @ I2 @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( member_set_set_int @ A @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( image_3513010637850279041et_int @ H @ ( a_r_co7378703583274672048t_unit @ R2 @ I2 @ A ) )
= ( a_r_co692709266861932262t_unit @ S @ ( image_3513010637850279041et_int @ H @ I2 ) @ ( H @ A ) ) ) ) ) ) ).
% a_coset_hom(2)
thf(fact_675_a__coset__hom_I2_J,axiom,
! [H: nat > int,R2: partia4692342223508353374t_unit,S: partia2818514838349642498t_unit,I2: set_nat,A: nat] :
( ( member_nat_int @ H @ ( ring_h1528583123504617310t_unit @ R2 @ S ) )
=> ( ( ord_less_eq_set_nat @ I2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( image_nat_int @ H @ ( a_r_co9089071853028257912t_unit @ R2 @ I2 @ A ) )
= ( a_r_co6205493800230438172t_unit @ S @ ( image_nat_int @ H @ I2 ) @ ( H @ A ) ) ) ) ) ) ).
% a_coset_hom(2)
thf(fact_676_a__coset__hom_I2_J,axiom,
! [H: nat > set_int,R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,I2: set_nat,A: nat] :
( ( member_nat_set_int @ H @ ( ring_h4752909569380436264t_unit @ R2 @ S ) )
=> ( ( ord_less_eq_set_nat @ I2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( image_nat_set_int @ H @ ( a_r_co9089071853028257912t_unit @ R2 @ I2 @ A ) )
= ( a_r_co692709266861932262t_unit @ S @ ( image_nat_set_int @ H @ I2 ) @ ( H @ A ) ) ) ) ) ) ).
% a_coset_hom(2)
thf(fact_677_set__add__hom,axiom,
! [H: set_int > int,R2: partia4934656038542163276t_unit,S: partia2818514838349642498t_unit,I2: set_set_int,J: set_set_int] :
( ( member_set_int_int @ H @ ( ring_h6167640903668269680t_unit @ R2 @ S ) )
=> ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( ord_le4403425263959731960et_int @ J @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( image_set_int_int @ H @ ( set_ad273131178244904872t_unit @ R2 @ I2 @ J ) )
= ( set_ad6660713454305125854t_unit @ S @ ( image_set_int_int @ H @ I2 ) @ ( image_set_int_int @ H @ J ) ) ) ) ) ) ).
% set_add_hom
thf(fact_678_set__add__hom,axiom,
! [H: set_int > set_int,R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,I2: set_set_int,J: set_set_int] :
( ( member5205197933313416826et_int @ H @ ( ring_h3404898052528352314t_unit @ R2 @ S ) )
=> ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( ord_le4403425263959731960et_int @ J @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( image_524474410958335435et_int @ H @ ( set_ad273131178244904872t_unit @ R2 @ I2 @ J ) )
= ( set_ad273131178244904872t_unit @ S @ ( image_524474410958335435et_int @ H @ I2 ) @ ( image_524474410958335435et_int @ H @ J ) ) ) ) ) ) ).
% set_add_hom
thf(fact_679_set__add__hom,axiom,
! [H: int > int,R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,I2: set_int,J: set_int] :
( ( member_int_int @ H @ ( ring_h9157551604493154234t_unit @ R2 @ S ) )
=> ( ( ord_less_eq_set_int @ I2 @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( ord_less_eq_set_int @ J @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( image_int_int @ H @ ( set_ad6660713454305125854t_unit @ R2 @ I2 @ J ) )
= ( set_ad6660713454305125854t_unit @ S @ ( image_int_int @ H @ I2 ) @ ( image_int_int @ H @ J ) ) ) ) ) ) ).
% set_add_hom
thf(fact_680_set__add__hom,axiom,
! [H: int > set_int,R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,I2: set_int,J: set_int] :
( ( member_int_set_int @ H @ ( ring_h1577560673974712708t_unit @ R2 @ S ) )
=> ( ( ord_less_eq_set_int @ I2 @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( ord_less_eq_set_int @ J @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( image_int_set_int @ H @ ( set_ad6660713454305125854t_unit @ R2 @ I2 @ J ) )
= ( set_ad273131178244904872t_unit @ S @ ( image_int_set_int @ H @ I2 ) @ ( image_int_set_int @ H @ J ) ) ) ) ) ) ).
% set_add_hom
thf(fact_681_set__add__hom,axiom,
! [H: set_set_int > int,R2: partia3601206958761670294t_unit,S: partia2818514838349642498t_unit,I2: set_set_set_int,J: set_set_set_int] :
( ( member2922251196090340986nt_int @ H @ ( ring_h8188925573458224678t_unit @ R2 @ S ) )
=> ( ( ord_le4317611570275147438et_int @ I2 @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( ord_le4317611570275147438et_int @ J @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( image_4671129442152397515nt_int @ H @ ( set_ad1658175378941472882t_unit @ R2 @ I2 @ J ) )
= ( set_ad6660713454305125854t_unit @ S @ ( image_4671129442152397515nt_int @ H @ I2 ) @ ( image_4671129442152397515nt_int @ H @ J ) ) ) ) ) ) ).
% set_add_hom
thf(fact_682_set__add__hom,axiom,
! [H: set_set_int > set_int,R2: partia3601206958761670294t_unit,S: partia4934656038542163276t_unit,I2: set_set_set_int,J: set_set_set_int] :
( ( member2198268278526348592et_int @ H @ ( ring_h8633152772399380464t_unit @ R2 @ S ) )
=> ( ( ord_le4317611570275147438et_int @ I2 @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( ord_le4317611570275147438et_int @ J @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( image_3513010637850279041et_int @ H @ ( set_ad1658175378941472882t_unit @ R2 @ I2 @ J ) )
= ( set_ad273131178244904872t_unit @ S @ ( image_3513010637850279041et_int @ H @ I2 ) @ ( image_3513010637850279041et_int @ H @ J ) ) ) ) ) ) ).
% set_add_hom
thf(fact_683_set__add__hom,axiom,
! [H: nat > int,R2: partia4692342223508353374t_unit,S: partia2818514838349642498t_unit,I2: set_nat,J: set_nat] :
( ( member_nat_int @ H @ ( ring_h1528583123504617310t_unit @ R2 @ S ) )
=> ( ( ord_less_eq_set_nat @ I2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( ord_less_eq_set_nat @ J @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( image_nat_int @ H @ ( set_ad320919470248169786t_unit @ R2 @ I2 @ J ) )
= ( set_ad6660713454305125854t_unit @ S @ ( image_nat_int @ H @ I2 ) @ ( image_nat_int @ H @ J ) ) ) ) ) ) ).
% set_add_hom
thf(fact_684_set__add__hom,axiom,
! [H: nat > set_int,R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,I2: set_nat,J: set_nat] :
( ( member_nat_set_int @ H @ ( ring_h4752909569380436264t_unit @ R2 @ S ) )
=> ( ( ord_less_eq_set_nat @ I2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( ord_less_eq_set_nat @ J @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( image_nat_set_int @ H @ ( set_ad320919470248169786t_unit @ R2 @ I2 @ J ) )
= ( set_ad273131178244904872t_unit @ S @ ( image_nat_set_int @ H @ I2 ) @ ( image_nat_set_int @ H @ J ) ) ) ) ) ) ).
% set_add_hom
thf(fact_685_a__coset__hom_I1_J,axiom,
! [H: set_int > int,R2: partia4934656038542163276t_unit,S: partia2818514838349642498t_unit,I2: set_set_int,A: set_int] :
( ( member_set_int_int @ H @ ( ring_h6167640903668269680t_unit @ R2 @ S ) )
=> ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( image_set_int_int @ H @ ( a_l_co3504123944629134560t_unit @ R2 @ A @ I2 ) )
= ( a_l_co457318074463736598t_unit @ S @ ( H @ A ) @ ( image_set_int_int @ H @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_686_a__coset__hom_I1_J,axiom,
! [H: set_int > set_int,R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,I2: set_set_int,A: set_int] :
( ( member5205197933313416826et_int @ H @ ( ring_h3404898052528352314t_unit @ R2 @ S ) )
=> ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( image_524474410958335435et_int @ H @ ( a_l_co3504123944629134560t_unit @ R2 @ A @ I2 ) )
= ( a_l_co3504123944629134560t_unit @ S @ ( H @ A ) @ ( image_524474410958335435et_int @ H @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_687_a__coset__hom_I1_J,axiom,
! [H: int > int,R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,I2: set_int,A: int] :
( ( member_int_int @ H @ ( ring_h9157551604493154234t_unit @ R2 @ S ) )
=> ( ( ord_less_eq_set_int @ I2 @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( member_int @ A @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( image_int_int @ H @ ( a_l_co457318074463736598t_unit @ R2 @ A @ I2 ) )
= ( a_l_co457318074463736598t_unit @ S @ ( H @ A ) @ ( image_int_int @ H @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_688_a__coset__hom_I1_J,axiom,
! [H: int > set_int,R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,I2: set_int,A: int] :
( ( member_int_set_int @ H @ ( ring_h1577560673974712708t_unit @ R2 @ S ) )
=> ( ( ord_less_eq_set_int @ I2 @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( member_int @ A @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( image_int_set_int @ H @ ( a_l_co457318074463736598t_unit @ R2 @ A @ I2 ) )
= ( a_l_co3504123944629134560t_unit @ S @ ( H @ A ) @ ( image_int_set_int @ H @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_689_a__coset__hom_I1_J,axiom,
! [H: set_set_int > int,R2: partia3601206958761670294t_unit,S: partia2818514838349642498t_unit,I2: set_set_set_int,A: set_set_int] :
( ( member2922251196090340986nt_int @ H @ ( ring_h8188925573458224678t_unit @ R2 @ S ) )
=> ( ( ord_le4317611570275147438et_int @ I2 @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( member_set_set_int @ A @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( image_4671129442152397515nt_int @ H @ ( a_l_co4050093972498570666t_unit @ R2 @ A @ I2 ) )
= ( a_l_co457318074463736598t_unit @ S @ ( H @ A ) @ ( image_4671129442152397515nt_int @ H @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_690_a__coset__hom_I1_J,axiom,
! [H: set_set_int > set_int,R2: partia3601206958761670294t_unit,S: partia4934656038542163276t_unit,I2: set_set_set_int,A: set_set_int] :
( ( member2198268278526348592et_int @ H @ ( ring_h8633152772399380464t_unit @ R2 @ S ) )
=> ( ( ord_le4317611570275147438et_int @ I2 @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( member_set_set_int @ A @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( image_3513010637850279041et_int @ H @ ( a_l_co4050093972498570666t_unit @ R2 @ A @ I2 ) )
= ( a_l_co3504123944629134560t_unit @ S @ ( H @ A ) @ ( image_3513010637850279041et_int @ H @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_691_a__coset__hom_I1_J,axiom,
! [H: nat > int,R2: partia4692342223508353374t_unit,S: partia2818514838349642498t_unit,I2: set_nat,A: nat] :
( ( member_nat_int @ H @ ( ring_h1528583123504617310t_unit @ R2 @ S ) )
=> ( ( ord_less_eq_set_nat @ I2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( image_nat_int @ H @ ( a_l_co3340896127261556338t_unit @ R2 @ A @ I2 ) )
= ( a_l_co457318074463736598t_unit @ S @ ( H @ A ) @ ( image_nat_int @ H @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_692_a__coset__hom_I1_J,axiom,
! [H: nat > set_int,R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,I2: set_nat,A: nat] :
( ( member_nat_set_int @ H @ ( ring_h4752909569380436264t_unit @ R2 @ S ) )
=> ( ( ord_less_eq_set_nat @ I2 @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( member_nat @ A @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( image_nat_set_int @ H @ ( a_l_co3340896127261556338t_unit @ R2 @ A @ I2 ) )
= ( a_l_co3504123944629134560t_unit @ S @ ( H @ A ) @ ( image_nat_set_int @ H @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_693_ring_Oring__ideal__imp__quot__ideal,axiom,
! [R2: partia3601206958761670294t_unit,I2: set_set_set_int,J: set_set_set_int] :
( ( ring_s5321399439445376583t_unit @ R2 )
=> ( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
=> ( ( ideal_1992573625310827463t_unit @ J @ R2 )
=> ( ideal_6096881031695155089t_unit @ ( image_7888150899047434989et_int @ ( a_r_co7378703583274672048t_unit @ R2 @ I2 ) @ J ) @ ( factRi2699865449404500690t_unit @ R2 @ I2 ) ) ) ) ) ).
% ring.ring_ideal_imp_quot_ideal
thf(fact_694_ring_Oring__ideal__imp__quot__ideal,axiom,
! [R2: partia2818514838349642498t_unit,I2: set_int,J: set_int] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( ( ideal_6787631597145370931t_unit @ I2 @ R2 )
=> ( ( ideal_6787631597145370931t_unit @ J @ R2 )
=> ( ideal_7262958097527394045t_unit @ ( image_int_set_int @ ( a_r_co6205493800230438172t_unit @ R2 @ I2 ) @ J ) @ ( factRi5755170488246124606t_unit @ R2 @ I2 ) ) ) ) ) ).
% ring.ring_ideal_imp_quot_ideal
thf(fact_695_ring_Oring__ideal__imp__quot__ideal,axiom,
! [R2: partia4934656038542163276t_unit,I2: set_set_int,J: set_set_int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
=> ( ( ideal_7262958097527394045t_unit @ J @ R2 )
=> ( ideal_1992573625310827463t_unit @ ( image_1010086626112315521et_int @ ( a_r_co692709266861932262t_unit @ R2 @ I2 ) @ J ) @ ( factRi3149420076008518152t_unit @ R2 @ I2 ) ) ) ) ) ).
% ring.ring_ideal_imp_quot_ideal
thf(fact_696_ring__hom__ring_Oset__add__ker__hom_I2_J,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,H: int > set_int,I2: set_int] :
( ( ring_h4095663460228855494t_unit @ R2 @ S @ H )
=> ( ( ord_less_eq_set_int @ I2 @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( image_int_set_int @ H @ ( set_ad6660713454305125854t_unit @ R2 @ ( a_kern5243884610727609734t_unit @ R2 @ S @ H ) @ I2 ) )
= ( image_int_set_int @ H @ I2 ) ) ) ) ).
% ring_hom_ring.set_add_ker_hom(2)
thf(fact_697_ring__hom__ring_Oset__add__ker__hom_I2_J,axiom,
! [R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,H: int > int,I2: set_int] :
( ( ring_h1606527851073305852t_unit @ R2 @ S @ H )
=> ( ( ord_less_eq_set_int @ I2 @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( image_int_int @ H @ ( set_ad6660713454305125854t_unit @ R2 @ ( a_kern6030145918964829116t_unit @ R2 @ S @ H ) @ I2 ) )
= ( image_int_int @ H @ I2 ) ) ) ) ).
% ring_hom_ring.set_add_ker_hom(2)
thf(fact_698_ring__hom__ring_Oset__add__ker__hom_I1_J,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,H: int > set_int,I2: set_int] :
( ( ring_h4095663460228855494t_unit @ R2 @ S @ H )
=> ( ( ord_less_eq_set_int @ I2 @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( image_int_set_int @ H @ ( set_ad6660713454305125854t_unit @ R2 @ I2 @ ( a_kern5243884610727609734t_unit @ R2 @ S @ H ) ) )
= ( image_int_set_int @ H @ I2 ) ) ) ) ).
% ring_hom_ring.set_add_ker_hom(1)
thf(fact_699_ring__hom__ring_Oset__add__ker__hom_I1_J,axiom,
! [R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,H: int > int,I2: set_int] :
( ( ring_h1606527851073305852t_unit @ R2 @ S @ H )
=> ( ( ord_less_eq_set_int @ I2 @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( image_int_int @ H @ ( set_ad6660713454305125854t_unit @ R2 @ I2 @ ( a_kern6030145918964829116t_unit @ R2 @ S @ H ) ) )
= ( image_int_int @ H @ I2 ) ) ) ) ).
% ring_hom_ring.set_add_ker_hom(1)
thf(fact_700_ring__hom__ring_Oquot__mem,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,H: int > set_int,X3: set_int] :
( ( ring_h4095663460228855494t_unit @ R2 @ S @ H )
=> ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( a_kern5243884610727609734t_unit @ R2 @ S @ H ) ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ ( partia8426541738980984321t_unit @ R2 ) )
& ( X3
= ( a_r_co6205493800230438172t_unit @ R2 @ ( a_kern5243884610727609734t_unit @ R2 @ S @ H ) @ X4 ) ) ) ) ) ).
% ring_hom_ring.quot_mem
thf(fact_701_ring__hom__ring_Oquot__mem,axiom,
! [R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,H: int > int,X3: set_int] :
( ( ring_h1606527851073305852t_unit @ R2 @ S @ H )
=> ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( factRi5755170488246124606t_unit @ R2 @ ( a_kern6030145918964829116t_unit @ R2 @ S @ H ) ) ) )
=> ? [X4: int] :
( ( member_int @ X4 @ ( partia8426541738980984321t_unit @ R2 ) )
& ( X3
= ( a_r_co6205493800230438172t_unit @ R2 @ ( a_kern6030145918964829116t_unit @ R2 @ S @ H ) @ X4 ) ) ) ) ) ).
% ring_hom_ring.quot_mem
thf(fact_702_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_703_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_704_r_OboundD__carrier,axiom,
! [N: nat,F: nat > set_int,M4: nat] :
( ( bound_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ N @ F )
=> ( ( ord_less_nat @ N @ M4 )
=> ( member_set_int @ ( F @ M4 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.boundD_carrier
thf(fact_705_r_Oint__embed__diff,axiom,
! [X: int,Y: int] :
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( minus_minus_int @ X @ Y ) )
= ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X ) @ ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ).
% r.int_embed_diff
thf(fact_706_r_Oa__rcosetsI,axiom,
! [H2: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( member_set_set_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 @ X ) @ ( a_RCOS5559887075240879033t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 ) ) ) ) ).
% r.a_rcosetsI
thf(fact_707_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_708_r_Oadd__ideals,axiom,
! [I2: set_set_int,J: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ideal_7262958097527394045t_unit @ J @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ideal_7262958097527394045t_unit @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ J ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.add_ideals
thf(fact_709_r_Oset__add__closed,axiom,
! [A3: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A3 @ ( 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 ) ) @ A3 @ B2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.set_add_closed
thf(fact_710_r_Oset__add__comm,axiom,
! [I2: set_set_int,J: set_set_int] :
( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ J @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 @ J )
= ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ J @ I2 ) ) ) ) ).
% r.set_add_comm
thf(fact_711_r_Osetadd__subset__G,axiom,
! [H2: set_set_int,K: set_set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 @ K ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.setadd_subset_G
thf(fact_712_r_Oa__l__coset__subset__G,axiom,
! [H2: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ H2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.a_l_coset_subset_G
thf(fact_713_r_Oa__lcos__mult__one,axiom,
! [M: set_set_int] :
( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ M )
= M ) ) ).
% r.a_lcos_mult_one
thf(fact_714_r_Oa__lcos__m__assoc,axiom,
! [M: set_set_int,G: set_int,H: set_int] :
( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ G @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ G @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H @ M ) )
= ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ G @ H ) @ M ) ) ) ) ) ).
% r.a_lcos_m_assoc
thf(fact_715_r_Oa__rcos__assoc__lcos,axiom,
! [H2: set_set_int,K: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K @ ( 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 ) ) @ H2 @ X ) @ K )
= ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ K ) ) ) ) ) ) ).
% r.a_rcos_assoc_lcos
thf(fact_716_r_Oa__setmult__rcos__assoc,axiom,
! [H2: set_set_int,K: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ K @ X ) )
= ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 @ K ) @ X ) ) ) ) ) ).
% r.a_setmult_rcos_assoc
thf(fact_717_r_Ogenideal__self,axiom,
! [S: set_set_int] :
( ( ord_le4403425263959731960et_int @ S @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_le4403425263959731960et_int @ S @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S ) ) ) ).
% r.genideal_self
thf(fact_718_r_Osubset__Idl__subset,axiom,
! [I2: set_set_int,H2: set_set_int] :
( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ H2 @ I2 )
=> ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 ) @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I2 ) ) ) ) ).
% r.subset_Idl_subset
thf(fact_719_r_Ogenideal__minimal,axiom,
! [I2: set_set_int,S: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ord_le4403425263959731960et_int @ S @ I2 )
=> ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S ) @ I2 ) ) ) ).
% r.genideal_minimal
thf(fact_720_r_Ominus__eq,axiom,
! [X: set_int,Y: set_int] :
( ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) ) ).
% r.minus_eq
thf(fact_721_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_722_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_723_r_OIdl__subset__ideal,axiom,
! [I2: set_set_int,H2: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 ) @ I2 )
= ( ord_le4403425263959731960et_int @ H2 @ I2 ) ) ) ) ).
% r.Idl_subset_ideal
thf(fact_724_r_Ogenideal__ideal,axiom,
! [S: set_set_int] :
( ( ord_le4403425263959731960et_int @ S @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ideal_7262958097527394045t_unit @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ S ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.genideal_ideal
thf(fact_725_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_726_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_727_diff__self__eq__0,axiom,
! [M4: nat] :
( ( minus_minus_nat @ M4 @ M4 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_728_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_729_nat__add__left__cancel__le,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M4 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M4 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_730_diff__diff__left,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J3 ) @ K2 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J3 @ K2 ) ) ) ).
% diff_diff_left
thf(fact_731_r_Oset__add__zero,axiom,
! [A3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A3 @ ( 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 ) @ A3 )
= A3 ) ) ).
% r.set_add_zero
thf(fact_732_zero__less__diff,axiom,
! [N: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M4 ) )
= ( ord_less_nat @ M4 @ N ) ) ).
% zero_less_diff
thf(fact_733_diff__is__0__eq,axiom,
! [M4: nat,N: nat] :
( ( ( minus_minus_nat @ M4 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M4 @ N ) ) ).
% diff_is_0_eq
thf(fact_734_diff__is__0__eq_H,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( minus_minus_nat @ M4 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_735_Nat_Odiff__diff__right,axiom,
! [K2: nat,J3: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J3 @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J3 ) ) ) ).
% Nat.diff_diff_right
thf(fact_736_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J3: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_737_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J3: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J3 @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J3 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_738_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_739_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_740_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_741_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ( minus_minus_nat @ J3 @ I )
= K2 )
= ( J3
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_742_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J3: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J3 @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J3 @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_743_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N3: nat] :
? [K4: nat] :
( N3
= ( plus_plus_nat @ M6 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_744_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J3: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J3 ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J3 @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_745_trans__le__add2,axiom,
! [I: nat,J3: nat,M4: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M4 @ J3 ) ) ) ).
% trans_le_add2
thf(fact_746_trans__le__add1,axiom,
! [I: nat,J3: nat,M4: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J3 @ M4 ) ) ) ).
% trans_le_add1
thf(fact_747_nat__le__linear,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
| ( ord_less_eq_nat @ N @ M4 ) ) ).
% nat_le_linear
thf(fact_748_mult__le__mono2,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J3 ) ) ) ).
% mult_le_mono2
thf(fact_749_mult__le__mono1,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J3 @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_750_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J3: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J3 @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J3 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_751_diff__le__mono2,axiom,
! [M4: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M4 ) ) ) ).
% diff_le_mono2
thf(fact_752_mult__le__mono,axiom,
! [I: nat,J3: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J3 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_753_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_754_le__diff__conv,axiom,
! [J3: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J3 @ K2 ) @ I )
= ( ord_less_eq_nat @ J3 @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_755_diff__le__self,axiom,
! [M4: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ N ) @ M4 ) ).
% diff_le_self
thf(fact_756_diff__le__mono,axiom,
! [M4: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_757_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M4 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M4 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M4 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_758_add__le__mono1,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J3 @ K2 ) ) ) ).
% add_le_mono1
thf(fact_759_le__diff__iff,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M4 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M4 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_760_eq__diff__iff,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M4 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M4 @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M4 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_761_add__le__mono,axiom,
! [I: nat,J3: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ) ).
% add_le_mono
thf(fact_762_le__antisym,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( ord_less_eq_nat @ N @ M4 )
=> ( M4 = N ) ) ) ).
% le_antisym
thf(fact_763_le__square,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ).
% le_square
thf(fact_764_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K2 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_765_eq__imp__le,axiom,
! [M4: nat,N: nat] :
( ( M4 = N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% eq_imp_le
thf(fact_766_le__trans,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_767_add__leD2,axiom,
! [M4: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_768_add__leD1,axiom,
! [M4: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K2 ) @ N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% add_leD1
thf(fact_769_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_770_le__cube,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ) ).
% le_cube
thf(fact_771_le__add2,axiom,
! [N: nat,M4: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M4 @ N ) ) ).
% le_add2
thf(fact_772_le__add1,axiom,
! [N: nat,M4: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M4 ) ) ).
% le_add1
thf(fact_773_add__leE,axiom,
! [M4: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M4 @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_774_add__mult__distrib,axiom,
! [M4: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M4 @ N ) @ K2 )
= ( plus_plus_nat @ ( times_times_nat @ M4 @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_775_add__mult__distrib2,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( times_times_nat @ K2 @ ( plus_plus_nat @ M4 @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K2 @ M4 ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_776_diff__mult__distrib2,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M4 @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M4 ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_777_diff__mult__distrib,axiom,
! [M4: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M4 @ N ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M4 @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_778_diff__commute,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J3 ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J3 ) ) ).
% diff_commute
thf(fact_779_diff__add__inverse2,axiom,
! [M4: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M4 @ N ) @ N )
= M4 ) ).
% diff_add_inverse2
thf(fact_780_diff__add__inverse,axiom,
! [N: nat,M4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M4 ) @ N )
= M4 ) ).
% diff_add_inverse
thf(fact_781_diff__cancel2,axiom,
! [M4: nat,K2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M4 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M4 @ N ) ) ).
% diff_cancel2
thf(fact_782_Nat_Odiff__cancel,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M4 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( minus_minus_nat @ M4 @ N ) ) ).
% Nat.diff_cancel
thf(fact_783_minus__nat_Odiff__0,axiom,
! [M4: nat] :
( ( minus_minus_nat @ M4 @ zero_zero_nat )
= M4 ) ).
% minus_nat.diff_0
thf(fact_784_diff__add__0,axiom,
! [N: nat,M4: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M4 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_785_diffs0__imp__equal,axiom,
! [M4: nat,N: nat] :
( ( ( minus_minus_nat @ M4 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M4 )
= zero_zero_nat )
=> ( M4 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_786_diff__less__mono2,axiom,
! [M4: nat,N: nat,L: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( ( ord_less_nat @ M4 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M4 ) ) ) ) ).
% diff_less_mono2
thf(fact_787_less__imp__diff__less,axiom,
! [J3: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J3 @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J3 @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_788_add__diff__inverse__nat,axiom,
! [M4: nat,N: nat] :
( ~ ( ord_less_nat @ M4 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M4 @ N ) )
= M4 ) ) ).
% add_diff_inverse_nat
thf(fact_789_less__diff__conv,axiom,
! [I: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J3 @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J3 ) ) ).
% less_diff_conv
thf(fact_790_less__diff__iff,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M4 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M4 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M4 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_791_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_792_less__diff__conv2,axiom,
! [K2: nat,J3: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J3 @ K2 ) @ I )
= ( ord_less_nat @ J3 @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_793_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_794_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_795_diff__less,axiom,
! [N: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_nat @ ( minus_minus_nat @ M4 @ N ) @ M4 ) ) ) ).
% diff_less
thf(fact_796_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M6: nat,N3: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_797_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_798_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_799_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_800_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_801_ring_Oint__embed__diff,axiom,
! [R2: partia4934656038542163276t_unit,X: int,Y: int] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ring_i2743490682209504680t_unit @ R2 @ ( minus_minus_int @ X @ Y ) )
= ( a_minu5974516859897376926t_unit @ R2 @ ( ring_i2743490682209504680t_unit @ R2 @ X ) @ ( ring_i2743490682209504680t_unit @ R2 @ Y ) ) ) ) ).
% ring.int_embed_diff
thf(fact_802_ring_Oint__embed__diff,axiom,
! [R2: partia2818514838349642498t_unit,X: int,Y: int] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( ( ring_i2003929451149106142t_unit @ R2 @ ( minus_minus_int @ X @ Y ) )
= ( a_minu8091582481160558292t_unit @ R2 @ ( ring_i2003929451149106142t_unit @ R2 @ X ) @ ( ring_i2003929451149106142t_unit @ R2 @ Y ) ) ) ) ).
% ring.int_embed_diff
thf(fact_803_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_804_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_805_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_806_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_807_s_Oinj__iff__trivial__ker,axiom,
( ( inj_on_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 ) ) ) @ i ) @ top_top_set_int )
= ( ( a_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) )
= ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ) ).
% s.inj_iff_trivial_ker
thf(fact_808_s_Otrivial__ker__imp__inj,axiom,
( ( ( a_kern5243884610727609734t_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 ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i ) )
= ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
=> ( inj_on_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 ) ) ) @ i ) @ top_top_set_int ) ) ).
% s.trivial_ker_imp_inj
thf(fact_809_r_Ochar__bound_I1_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_less_eq_nat @ ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ X ) ) ) ).
% r.char_bound(1)
thf(fact_810_r_Oembed__char__eq__0,axiom,
( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( semiri1314217659103216013at_int @ ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.embed_char_eq_0
thf(fact_811_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_812_r_Ochar__bound_I2_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i2743490682209504680t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).
% r.char_bound(2)
thf(fact_813_negative__eq__positive,axiom,
! [N: nat,M4: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M4 ) )
= ( ( N = zero_zero_nat )
& ( M4 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_814_negative__zle,axiom,
! [N: nat,M4: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M4 ) ) ).
% negative_zle
thf(fact_815_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_816_inj__on__image,axiom,
! [F: int > set_int,A3: set_set_int] :
( ( inj_on_int_set_int @ F @ ( comple3221217463730067765et_int @ A3 ) )
=> ( inj_on6285404204842837485et_int @ ( image_int_set_int @ F ) @ A3 ) ) ).
% inj_on_image
thf(fact_817_inj__on__image,axiom,
! [F: int > int,A3: set_set_int] :
( ( inj_on_int_int @ F @ ( comple3221217463730067765et_int @ A3 ) )
=> ( inj_on6435365835345961783et_int @ ( image_int_int @ F ) @ A3 ) ) ).
% inj_on_image
thf(fact_818_inj__on__image,axiom,
! [F: set_int > set_set_int,A3: set_set_set_int] :
( ( inj_on6285404204842837485et_int @ F @ ( comple7281953568134767595et_int @ A3 ) )
=> ( inj_on825803521015836761et_int @ ( image_1010086626112315521et_int @ F ) @ A3 ) ) ).
% inj_on_image
thf(fact_819_inj__on__image,axiom,
! [F: set_int > set_int,A3: set_set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ ( comple7281953568134767595et_int @ A3 ) )
=> ( inj_on6097848081577782435et_int @ ( image_524474410958335435et_int @ F ) @ A3 ) ) ).
% inj_on_image
thf(fact_820_ring__hom__ring_Ochar__consistent,axiom,
! [R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,H: set_int > set_int] :
( ( ring_h98094481560215932t_unit @ R2 @ S @ H )
=> ( ( inj_on6435365835345961783et_int @ H @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( ring_c6147214092195050492t_unit @ R2 )
= ( ring_c6147214092195050492t_unit @ S ) ) ) ) ).
% ring_hom_ring.char_consistent
thf(fact_821_ring__hom__ring_Ochar__consistent,axiom,
! [R2: partia4934656038542163276t_unit,S: partia2818514838349642498t_unit,H: set_int > int] :
( ( ring_h8685743689922412466t_unit @ R2 @ S @ H )
=> ( ( inj_on_set_int_int @ H @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ( ring_c6147214092195050492t_unit @ R2 )
= ( ring_c8451697193457390130t_unit @ S ) ) ) ) ).
% ring_hom_ring.char_consistent
thf(fact_822_ring__hom__ring_Ochar__consistent,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,H: int > set_int] :
( ( ring_h4095663460228855494t_unit @ R2 @ S @ H )
=> ( ( inj_on_int_set_int @ H @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( ring_c8451697193457390130t_unit @ R2 )
= ( ring_c6147214092195050492t_unit @ S ) ) ) ) ).
% ring_hom_ring.char_consistent
thf(fact_823_ring__hom__ring_Ochar__consistent,axiom,
! [R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,H: int > int] :
( ( ring_h1606527851073305852t_unit @ R2 @ S @ H )
=> ( ( inj_on_int_int @ H @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ( ring_c8451697193457390130t_unit @ R2 )
= ( ring_c8451697193457390130t_unit @ S ) ) ) ) ).
% ring_hom_ring.char_consistent
thf(fact_824_ring__hom__ring_Ochar__consistent,axiom,
! [R2: partia3601206958761670294t_unit,S: partia4934656038542163276t_unit,H: set_set_int > set_int] :
( ( ring_h8546004371043296562t_unit @ R2 @ S @ H )
=> ( ( inj_on8788328216580801005et_int @ H @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( ring_c554216831298059462t_unit @ R2 )
= ( ring_c6147214092195050492t_unit @ S ) ) ) ) ).
% ring_hom_ring.char_consistent
thf(fact_825_ring__hom__ring_Ochar__consistent,axiom,
! [R2: partia3601206958761670294t_unit,S: partia2818514838349642498t_unit,H: set_set_int > int] :
( ( ring_h4882122002490088296t_unit @ R2 @ S @ H )
=> ( ( inj_on1358648829685248055nt_int @ H @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ( ring_c554216831298059462t_unit @ R2 )
= ( ring_c8451697193457390130t_unit @ S ) ) ) ) ).
% ring_hom_ring.char_consistent
thf(fact_826_ring__hom__ring_Ochar__consistent,axiom,
! [R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,H: nat > set_int] :
( ( ring_h7271012355634579050t_unit @ R2 @ S @ H )
=> ( ( inj_on_nat_set_int @ H @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( ring_c2111903209400434062t_unit @ R2 )
= ( ring_c6147214092195050492t_unit @ S ) ) ) ) ).
% ring_hom_ring.char_consistent
thf(fact_827_ring__hom__ring_Ochar__consistent,axiom,
! [R2: partia4692342223508353374t_unit,S: partia2818514838349642498t_unit,H: nat > int] :
( ( ring_h3200931406939544736t_unit @ R2 @ S @ H )
=> ( ( inj_on_nat_int @ H @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ( ring_c2111903209400434062t_unit @ R2 )
= ( ring_c8451697193457390130t_unit @ S ) ) ) ) ).
% ring_hom_ring.char_consistent
thf(fact_828_int__int__eq,axiom,
! [M4: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M4 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M4 = N ) ) ).
% int_int_eq
thf(fact_829_ring_Oembed__char__eq__0,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ring_i2743490682209504680t_unit @ R2 @ ( semiri1314217659103216013at_int @ ( ring_c6147214092195050492t_unit @ R2 ) ) )
= ( zero_s6269048424454532197t_unit @ R2 ) ) ) ).
% ring.embed_char_eq_0
thf(fact_830_ring_Oembed__char__eq__0,axiom,
! [R2: partia2818514838349642498t_unit] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( ( ring_i2003929451149106142t_unit @ R2 @ ( semiri1314217659103216013at_int @ ( ring_c8451697193457390130t_unit @ R2 ) ) )
= ( zero_i2266321264637750939t_unit @ R2 ) ) ) ).
% ring.embed_char_eq_0
thf(fact_831_ring__hom__ring_Oinj__iff__trivial__ker,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,H: int > set_int] :
( ( ring_h4095663460228855494t_unit @ R2 @ S @ H )
=> ( ( inj_on_int_set_int @ H @ ( partia8426541738980984321t_unit @ R2 ) )
= ( ( a_kern5243884610727609734t_unit @ R2 @ S @ H )
= ( insert_int @ ( zero_i2266321264637750939t_unit @ R2 ) @ bot_bot_set_int ) ) ) ) ).
% ring_hom_ring.inj_iff_trivial_ker
thf(fact_832_ring__hom__ring_Oinj__iff__trivial__ker,axiom,
! [R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,H: int > int] :
( ( ring_h1606527851073305852t_unit @ R2 @ S @ H )
=> ( ( inj_on_int_int @ H @ ( partia8426541738980984321t_unit @ R2 ) )
= ( ( a_kern6030145918964829116t_unit @ R2 @ S @ H )
= ( insert_int @ ( zero_i2266321264637750939t_unit @ R2 ) @ bot_bot_set_int ) ) ) ) ).
% ring_hom_ring.inj_iff_trivial_ker
thf(fact_833_ring__hom__ring_Otrivial__ker__imp__inj,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,H: int > set_int] :
( ( ring_h4095663460228855494t_unit @ R2 @ S @ H )
=> ( ( ( a_kern5243884610727609734t_unit @ R2 @ S @ H )
= ( insert_int @ ( zero_i2266321264637750939t_unit @ R2 ) @ bot_bot_set_int ) )
=> ( inj_on_int_set_int @ H @ ( partia8426541738980984321t_unit @ R2 ) ) ) ) ).
% ring_hom_ring.trivial_ker_imp_inj
thf(fact_834_ring__hom__ring_Otrivial__ker__imp__inj,axiom,
! [R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,H: int > int] :
( ( ring_h1606527851073305852t_unit @ R2 @ S @ H )
=> ( ( ( a_kern6030145918964829116t_unit @ R2 @ S @ H )
= ( insert_int @ ( zero_i2266321264637750939t_unit @ R2 ) @ bot_bot_set_int ) )
=> ( inj_on_int_int @ H @ ( partia8426541738980984321t_unit @ R2 ) ) ) ) ).
% ring_hom_ring.trivial_ker_imp_inj
thf(fact_835_ring_Ofinite__carr__imp__char__ge__0,axiom,
! [R2: partia4934656038542163276t_unit] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ R2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c6147214092195050492t_unit @ R2 ) ) ) ) ).
% ring.finite_carr_imp_char_ge_0
thf(fact_836_ring_Ofinite__carr__imp__char__ge__0,axiom,
! [R2: partia2818514838349642498t_unit] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( ( finite_finite_int @ ( partia8426541738980984321t_unit @ R2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c8451697193457390130t_unit @ R2 ) ) ) ) ).
% ring.finite_carr_imp_char_ge_0
thf(fact_837_ring_Ofinite__carr__imp__char__ge__0,axiom,
! [R2: partia3601206958761670294t_unit] :
( ( ring_s5321399439445376583t_unit @ R2 )
=> ( ( finite4249678464180374575et_int @ ( partia1981568256555877781t_unit @ R2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c554216831298059462t_unit @ R2 ) ) ) ) ).
% ring.finite_carr_imp_char_ge_0
thf(fact_838_ring_Ofinite__carr__imp__char__ge__0,axiom,
! [R2: partia4692342223508353374t_unit] :
( ( ring_n9194430563101542159t_unit @ R2 )
=> ( ( finite_finite_nat @ ( partia3499330772048238685t_unit @ R2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c2111903209400434062t_unit @ R2 ) ) ) ) ).
% ring.finite_carr_imp_char_ge_0
thf(fact_839_plus__int__code_I1_J,axiom,
! [K2: int] :
( ( plus_plus_int @ K2 @ zero_zero_int )
= K2 ) ).
% plus_int_code(1)
thf(fact_840_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_841_times__int__code_I1_J,axiom,
! [K2: int] :
( ( times_times_int @ K2 @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_842_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_843_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_844_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_845_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_846_int__cases2,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_847_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_848_int__diff__cases,axiom,
! [Z: int] :
~ ! [M7: nat,N2: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M7 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_849_minus__int__code_I1_J,axiom,
! [K2: int] :
( ( minus_minus_int @ K2 @ zero_zero_int )
= K2 ) ).
% minus_int_code(1)
thf(fact_850_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_851_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_852_infinite__UNIV__int,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_int
thf(fact_853_ring_Ochar__bound_I2_J,axiom,
! [R2: partia4934656038542163276t_unit,X: nat] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i2743490682209504680t_unit @ R2 @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_s6269048424454532197t_unit @ R2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c6147214092195050492t_unit @ R2 ) ) ) ) ) ).
% ring.char_bound(2)
thf(fact_854_ring_Ochar__bound_I2_J,axiom,
! [R2: partia2818514838349642498t_unit,X: nat] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i2003929451149106142t_unit @ R2 @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_i2266321264637750939t_unit @ R2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c8451697193457390130t_unit @ R2 ) ) ) ) ) ).
% ring.char_bound(2)
thf(fact_855_ring_Ochar__bound_I1_J,axiom,
! [R2: partia4934656038542163276t_unit,X: nat] :
( ( ring_s5316885176909347197t_unit @ R2 )
=> ( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i2743490682209504680t_unit @ R2 @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_s6269048424454532197t_unit @ R2 ) )
=> ( ord_less_eq_nat @ ( ring_c6147214092195050492t_unit @ R2 ) @ X ) ) ) ) ).
% ring.char_bound(1)
thf(fact_856_ring_Ochar__bound_I1_J,axiom,
! [R2: partia2818514838349642498t_unit,X: nat] :
( ( ring_i6310852510303722419t_unit @ R2 )
=> ( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i2003929451149106142t_unit @ R2 @ ( semiri1314217659103216013at_int @ X ) )
= ( zero_i2266321264637750939t_unit @ R2 ) )
=> ( ord_less_eq_nat @ ( ring_c8451697193457390130t_unit @ R2 ) @ X ) ) ) ) ).
% ring.char_bound(1)
thf(fact_857_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_858_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ? [N2: nat] :
( K2
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_859_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ~ ! [N2: nat] :
( K2
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_860_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_861_zadd__int__left,axiom,
! [M4: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M4 @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_862_int__ge__induct,axiom,
! [K2: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K2 @ I )
=> ( ( P @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_863_zmult__zless__mono2,axiom,
! [I: int,J3: int,K2: int] :
( ( ord_less_int @ I @ J3 )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ord_less_int @ ( times_times_int @ K2 @ I ) @ ( times_times_int @ K2 @ J3 ) ) ) ) ).
% zmult_zless_mono2
thf(fact_864_pos__zmult__eq__1__iff__lemma,axiom,
! [M4: int,N: int] :
( ( ( times_times_int @ M4 @ N )
= one_one_int )
=> ( ( M4 = one_one_int )
| ( M4
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_865_zmult__eq__1__iff,axiom,
! [M4: int,N: int] :
( ( ( times_times_int @ M4 @ N )
= one_one_int )
= ( ( ( M4 = one_one_int )
& ( N = one_one_int ) )
| ( ( M4
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_866_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_867_int__gr__induct,axiom,
! [K2: int,I: int,P: int > $o] :
( ( ord_less_int @ K2 @ I )
=> ( ( P @ ( plus_plus_int @ K2 @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K2 @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_868_zle__int,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M4 @ N ) ) ).
% zle_int
thf(fact_869_not__int__zless__negative,axiom,
! [N: nat,M4: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M4 ) ) ) ).
% not_int_zless_negative
thf(fact_870_int__le__induct,axiom,
! [I: int,K2: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K2 )
=> ( ( P @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K2 )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_871_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_872_int__less__induct,axiom,
! [I: int,K2: int,P: int > $o] :
( ( ord_less_int @ I @ K2 )
=> ( ( P @ ( minus_minus_int @ K2 @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K2 )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_873_int__a__minus__eq,axiom,
! [X: int,Y: int] :
( ( a_minu8091582481160558292t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ Y )
= ( minus_minus_int @ X @ Y ) ) ).
% int_a_minus_eq
thf(fact_874_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_875_pos__zmult__eq__1__iff,axiom,
! [M4: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M4 )
=> ( ( ( times_times_int @ M4 @ N )
= one_one_int )
= ( ( M4 = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_876_nonpos__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ~ ! [N2: nat] :
( K2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_877_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_878_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_879_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_880_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_881_int__cases4,axiom,
! [M4: int] :
( ! [N2: nat] :
( M4
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M4
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_882_int__zle__neg,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M4 ) ) )
= ( ( N = zero_zero_nat )
& ( M4 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_883_int__induct,axiom,
! [P: int > $o,K2: int,I: int] :
( ( P @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K2 )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_884_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_885_int__cases3,axiom,
! [K2: int] :
( ( K2 != zero_zero_int )
=> ( ! [N2: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_886_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N2: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_887_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K2
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_888_zmult__zless__mono2__lemma,axiom,
! [I: int,J3: int,K2: nat] :
( ( ord_less_int @ I @ J3 )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ J3 ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_889_neg__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ K2 @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_890_s_Onon__trivial__field__hom__imp__inj,axiom,
( ( 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 ) ) ) )
=> ( ( ( 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 ) ) ) @ i ) @ top_top_set_int )
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ bot_bot_set_set_int ) )
=> ( inj_on_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 ) ) ) @ i ) @ top_top_set_int ) ) ) ).
% s.non_trivial_field_hom_imp_inj
thf(fact_891_s_Ochar__consistent,axiom,
( ( inj_on_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 ) ) ) @ i ) @ top_top_set_int )
=> ( ( ring_c8451697193457390130t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) )
= ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% s.char_consistent
thf(fact_892_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_893_r_Oadd_Oinv__inj,axiom,
inj_on6435365835345961783et_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).
% r.add.inv_inj
thf(fact_894_r_Ofinite__carr__imp__char__ge__0,axiom,
( ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c6147214092195050492t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.finite_carr_imp_char_ge_0
thf(fact_895_i_Oadd_Oinv__inj,axiom,
inj_on_int_int @ ( a_inv_8811962894454695315t_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 ) ) ) ) @ ( 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 ) ) ) ) ).
% i.add.inv_inj
thf(fact_896_inj__of__nat,axiom,
inj_on_nat_int @ semiri1314217659103216013at_int @ top_top_set_nat ).
% inj_of_nat
thf(fact_897_non__trivial__field__hom__is__inj,axiom,
! [H: set_int > int,R2: partia4934656038542163276t_unit,S: partia2818514838349642498t_unit] :
( ( member_set_int_int @ H @ ( ring_h6167640903668269680t_unit @ R2 @ S ) )
=> ( ( field_5943785737635511755t_unit @ R2 )
=> ( ( field_5117527561578272769t_unit @ S )
=> ( inj_on_set_int_int @ H @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% non_trivial_field_hom_is_inj
thf(fact_898_non__trivial__field__hom__is__inj,axiom,
! [H: set_int > set_int,R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit] :
( ( member5205197933313416826et_int @ H @ ( ring_h3404898052528352314t_unit @ R2 @ S ) )
=> ( ( field_5943785737635511755t_unit @ R2 )
=> ( ( field_5943785737635511755t_unit @ S )
=> ( inj_on6435365835345961783et_int @ H @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% non_trivial_field_hom_is_inj
thf(fact_899_non__trivial__field__hom__is__inj,axiom,
! [H: int > int,R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit] :
( ( member_int_int @ H @ ( ring_h9157551604493154234t_unit @ R2 @ S ) )
=> ( ( field_5117527561578272769t_unit @ R2 )
=> ( ( field_5117527561578272769t_unit @ S )
=> ( inj_on_int_int @ H @ ( partia8426541738980984321t_unit @ R2 ) ) ) ) ) ).
% non_trivial_field_hom_is_inj
thf(fact_900_non__trivial__field__hom__is__inj,axiom,
! [H: int > set_int,R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit] :
( ( member_int_set_int @ H @ ( ring_h1577560673974712708t_unit @ R2 @ S ) )
=> ( ( field_5117527561578272769t_unit @ R2 )
=> ( ( field_5943785737635511755t_unit @ S )
=> ( inj_on_int_set_int @ H @ ( partia8426541738980984321t_unit @ R2 ) ) ) ) ) ).
% non_trivial_field_hom_is_inj
thf(fact_901_non__trivial__field__hom__is__inj,axiom,
! [H: set_set_int > int,R2: partia3601206958761670294t_unit,S: partia2818514838349642498t_unit] :
( ( member2922251196090340986nt_int @ H @ ( ring_h8188925573458224678t_unit @ R2 @ S ) )
=> ( ( field_4969826516285655701t_unit @ R2 )
=> ( ( field_5117527561578272769t_unit @ S )
=> ( inj_on1358648829685248055nt_int @ H @ ( partia1981568256555877781t_unit @ R2 ) ) ) ) ) ).
% non_trivial_field_hom_is_inj
thf(fact_902_non__trivial__field__hom__is__inj,axiom,
! [H: set_set_int > set_int,R2: partia3601206958761670294t_unit,S: partia4934656038542163276t_unit] :
( ( member2198268278526348592et_int @ H @ ( ring_h8633152772399380464t_unit @ R2 @ S ) )
=> ( ( field_4969826516285655701t_unit @ R2 )
=> ( ( field_5943785737635511755t_unit @ S )
=> ( inj_on8788328216580801005et_int @ H @ ( partia1981568256555877781t_unit @ R2 ) ) ) ) ) ).
% non_trivial_field_hom_is_inj
thf(fact_903_non__trivial__field__hom__is__inj,axiom,
! [H: nat > int,R2: partia4692342223508353374t_unit,S: partia2818514838349642498t_unit] :
( ( member_nat_int @ H @ ( ring_h1528583123504617310t_unit @ R2 @ S ) )
=> ( ( field_8001105614376092509t_unit @ R2 )
=> ( ( field_5117527561578272769t_unit @ S )
=> ( inj_on_nat_int @ H @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% non_trivial_field_hom_is_inj
thf(fact_904_non__trivial__field__hom__is__inj,axiom,
! [H: nat > set_int,R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit] :
( ( member_nat_set_int @ H @ ( ring_h4752909569380436264t_unit @ R2 @ S ) )
=> ( ( field_8001105614376092509t_unit @ R2 )
=> ( ( field_5943785737635511755t_unit @ S )
=> ( inj_on_nat_set_int @ H @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% non_trivial_field_hom_is_inj
thf(fact_905_fin__zfact,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% fin_zfact
thf(fact_906_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
! [R2: partia4934656038542163276t_unit,S: partia2818514838349642498t_unit,H: set_int > int] :
( ( ring_h8685743689922412466t_unit @ R2 @ S @ H )
=> ( ( field_5943785737635511755t_unit @ R2 )
=> ( ( ( image_set_int_int @ H @ ( partia966996272515721803t_unit @ R2 ) )
!= ( insert_int @ ( zero_i2266321264637750939t_unit @ S ) @ bot_bot_set_int ) )
=> ( inj_on_set_int_int @ H @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_907_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
! [R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,H: set_int > set_int] :
( ( ring_h98094481560215932t_unit @ R2 @ S @ H )
=> ( ( field_5943785737635511755t_unit @ R2 )
=> ( ( ( image_524474410958335435et_int @ H @ ( partia966996272515721803t_unit @ R2 ) )
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ S ) @ bot_bot_set_set_int ) )
=> ( inj_on6435365835345961783et_int @ H @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).
% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_908_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
! [R2: partia2818514838349642498t_unit,S: partia2818514838349642498t_unit,H: int > int] :
( ( ring_h1606527851073305852t_unit @ R2 @ S @ H )
=> ( ( field_5117527561578272769t_unit @ R2 )
=> ( ( ( image_int_int @ H @ ( partia8426541738980984321t_unit @ R2 ) )
!= ( insert_int @ ( zero_i2266321264637750939t_unit @ S ) @ bot_bot_set_int ) )
=> ( inj_on_int_int @ H @ ( partia8426541738980984321t_unit @ R2 ) ) ) ) ) ).
% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_909_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
! [R2: partia2818514838349642498t_unit,S: partia4934656038542163276t_unit,H: int > set_int] :
( ( ring_h4095663460228855494t_unit @ R2 @ S @ H )
=> ( ( field_5117527561578272769t_unit @ R2 )
=> ( ( ( image_int_set_int @ H @ ( partia8426541738980984321t_unit @ R2 ) )
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ S ) @ bot_bot_set_set_int ) )
=> ( inj_on_int_set_int @ H @ ( partia8426541738980984321t_unit @ R2 ) ) ) ) ) ).
% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_910_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
! [R2: partia3601206958761670294t_unit,S: partia2818514838349642498t_unit,H: set_set_int > int] :
( ( ring_h4882122002490088296t_unit @ R2 @ S @ H )
=> ( ( field_4969826516285655701t_unit @ R2 )
=> ( ( ( image_4671129442152397515nt_int @ H @ ( partia1981568256555877781t_unit @ R2 ) )
!= ( insert_int @ ( zero_i2266321264637750939t_unit @ S ) @ bot_bot_set_int ) )
=> ( inj_on1358648829685248055nt_int @ H @ ( partia1981568256555877781t_unit @ R2 ) ) ) ) ) ).
% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_911_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
! [R2: partia3601206958761670294t_unit,S: partia4934656038542163276t_unit,H: set_set_int > set_int] :
( ( ring_h8546004371043296562t_unit @ R2 @ S @ H )
=> ( ( field_4969826516285655701t_unit @ R2 )
=> ( ( ( image_3513010637850279041et_int @ H @ ( partia1981568256555877781t_unit @ R2 ) )
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ S ) @ bot_bot_set_set_int ) )
=> ( inj_on8788328216580801005et_int @ H @ ( partia1981568256555877781t_unit @ R2 ) ) ) ) ) ).
% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_912_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
! [R2: partia4692342223508353374t_unit,S: partia2818514838349642498t_unit,H: nat > int] :
( ( ring_h3200931406939544736t_unit @ R2 @ S @ H )
=> ( ( field_8001105614376092509t_unit @ R2 )
=> ( ( ( image_nat_int @ H @ ( partia3499330772048238685t_unit @ R2 ) )
!= ( insert_int @ ( zero_i2266321264637750939t_unit @ S ) @ bot_bot_set_int ) )
=> ( inj_on_nat_int @ H @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_913_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
! [R2: partia4692342223508353374t_unit,S: partia4934656038542163276t_unit,H: nat > set_int] :
( ( ring_h7271012355634579050t_unit @ R2 @ S @ H )
=> ( ( field_8001105614376092509t_unit @ R2 )
=> ( ( ( image_nat_set_int @ H @ ( partia3499330772048238685t_unit @ R2 ) )
!= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ S ) @ bot_bot_set_set_int ) )
=> ( inj_on_nat_set_int @ H @ ( partia3499330772048238685t_unit @ R2 ) ) ) ) ) ).
% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_914_int_Ocring__fieldI2,axiom,
( ( zero_zero_int != one_one_int )
=> ( ! [A4: int] :
( ( member_int @ A4 @ top_top_set_int )
=> ( ( A4 != zero_zero_int )
=> ? [X2: int] :
( ( member_int @ X2 @ top_top_set_int )
& ( ( times_times_int @ A4 @ X2 )
= 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_915_int_Ogenideal__self,axiom,
! [S: set_int] :
( ( ord_less_eq_set_int @ S @ top_top_set_int )
=> ( ord_less_eq_set_int @ S @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ S ) ) ) ).
% int.genideal_self
thf(fact_916_int_Osubset__Idl__subset,axiom,
! [I2: set_int,H2: set_int] :
( ( ord_less_eq_set_int @ I2 @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ H2 @ 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 ) ) ) @ H2 ) @ ( 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_917_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_918_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_919_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_920_int_Ogenideal__ideal,axiom,
! [S: set_int] :
( ( ord_less_eq_set_int @ S @ 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 ) ) ) @ S ) @ ( 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_921_int_OIdl__subset__ideal,axiom,
! [I2: set_int,H2: 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 @ H2 @ 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 ) ) ) @ H2 ) @ I2 )
= ( ord_less_eq_set_int @ H2 @ I2 ) ) ) ) ).
% int.Idl_subset_ideal
thf(fact_922_int_Ogenideal__minimal,axiom,
! [I2: set_int,S: 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 @ S @ 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 ) ) ) @ S ) @ I2 ) ) ) ).
% int.genideal_minimal
thf(fact_923_int_Oembed__char__eq__0,axiom,
( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( semiri1314217659103216013at_int @ ( ring_c8451697193457390130t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) )
= zero_zero_int ) ).
% int.embed_char_eq_0
thf(fact_924_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_925_int__Idl__subset__ideal,axiom,
! [K2: 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 @ K2 @ 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 @ K2 @ ( 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_926_int_Ofinite__carr__imp__char__ge__0,axiom,
( ( finite_finite_int @ top_top_set_int )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c8451697193457390130t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ).
% int.finite_carr_imp_char_ge_0
thf(fact_927_ZFact__def,axiom,
( zFact
= ( ^ [K4: 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 @ K4 @ bot_bot_set_int ) ) ) ) ) ).
% ZFact_def
thf(fact_928_int_Ochar__bound_I2_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( semiri1314217659103216013at_int @ X ) )
= zero_zero_int )
=> ( ord_less_nat @ zero_zero_nat @ ( ring_c8451697193457390130t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ) ).
% int.char_bound(2)
thf(fact_929_int_Ochar__bound_I1_J,axiom,
! [X: nat] :
( ( ord_less_nat @ zero_zero_nat @ X )
=> ( ( ( ring_i2003929451149106142t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( semiri1314217659103216013at_int @ X ) )
= zero_zero_int )
=> ( ord_less_eq_nat @ ( ring_c8451697193457390130t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) @ X ) ) ) ).
% int.char_bound(1)
thf(fact_930_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_931_r_Onoetherian__ringI,axiom,
( ! [I5: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I5 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ? [A5: set_set_int] :
( ( ord_le4403425263959731960et_int @ A5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
& ( finite6197958912794628473et_int @ A5 )
& ( I5
= ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A5 ) ) ) )
=> ( ring_n4918880612521764245t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ).
% r.noetherian_ringI
thf(fact_932_r_Oorder__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_4716970363388151434t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
= ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).
% r.order_gt_0_iff_finite
thf(fact_933_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M4 ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M4 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_934_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M4 ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M4 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_935_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M4 )
= ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( M4 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_936_nat__mult__less__cancel1,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M4 ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_nat @ M4 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_937_nat__mult__eq__cancel1,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M4 )
= ( times_times_nat @ K2 @ N ) )
= ( M4 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_938_nat__mult__le__cancel1,axiom,
! [K2: nat,M4: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M4 ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M4 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_939_nat__eq__add__iff1,axiom,
! [J3: nat,I: nat,U: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ J3 @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 )
= ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J3 ) @ U ) @ M4 )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_940_nat__eq__add__iff2,axiom,
! [I: nat,J3: nat,U: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 )
= ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N ) )
= ( M4
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J3 @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_941_nat__le__add__iff1,axiom,
! [J3: nat,I: nat,U: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ J3 @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J3 ) @ U ) @ M4 ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_942_nat__le__add__iff2,axiom,
! [I: nat,J3: nat,U: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N ) )
= ( ord_less_eq_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J3 @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_943_nat__diff__add__eq1,axiom,
! [J3: nat,I: nat,U: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ J3 @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J3 ) @ U ) @ M4 ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_944_nat__diff__add__eq2,axiom,
! [I: nat,J3: nat,U: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N ) )
= ( minus_minus_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J3 @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_945_nat__less__add__iff1,axiom,
! [J3: nat,I: nat,U: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ J3 @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J3 ) @ U ) @ M4 ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_946_nat__less__add__iff2,axiom,
! [I: nat,J3: nat,U: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J3 @ U ) @ N ) )
= ( ord_less_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J3 @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_947_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_948_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_949_inj__on__insert,axiom,
! [F: int > int,A: int,A3: set_int] :
( ( inj_on_int_int @ F @ ( insert_int @ A @ A3 ) )
= ( ( inj_on_int_int @ F @ A3 )
& ~ ( member_int @ ( F @ A ) @ ( image_int_int @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_950_inj__on__insert,axiom,
! [F: int > set_int,A: int,A3: set_int] :
( ( inj_on_int_set_int @ F @ ( insert_int @ A @ A3 ) )
= ( ( inj_on_int_set_int @ F @ A3 )
& ~ ( member_set_int @ ( F @ A ) @ ( image_int_set_int @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_951_inj__on__insert,axiom,
! [F: int > int > set_int,A: int,A3: set_int] :
( ( inj_on8549566020547182028et_int @ F @ ( insert_int @ A @ A3 ) )
= ( ( inj_on8549566020547182028et_int @ F @ A3 )
& ~ ( member_int_set_int @ ( F @ A ) @ ( image_3526486554573990496et_int @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_952_inj__on__insert,axiom,
! [F: int > set_set_int,A: int,A3: set_int] :
( ( inj_on3305102505822124087et_int @ F @ ( insert_int @ A @ A3 ) )
= ( ( inj_on3305102505822124087et_int @ F @ A3 )
& ~ ( member_set_set_int @ ( F @ A ) @ ( image_6617583118289273547et_int @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_953_inj__on__insert,axiom,
! [F: int > nat,A: int,A3: set_int] :
( ( inj_on_int_nat @ F @ ( insert_int @ A @ A3 ) )
= ( ( inj_on_int_nat @ F @ A3 )
& ~ ( member_nat @ ( F @ A ) @ ( image_int_nat @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_954_inj__on__insert,axiom,
! [F: set_int > int,A: set_int,A3: set_set_int] :
( ( inj_on_set_int_int @ F @ ( insert_set_int @ A @ A3 ) )
= ( ( inj_on_set_int_int @ F @ A3 )
& ~ ( member_int @ ( F @ A ) @ ( image_set_int_int @ F @ ( minus_8897228262479074673et_int @ A3 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_955_inj__on__insert,axiom,
! [F: set_int > set_int,A: set_int,A3: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ ( insert_set_int @ A @ A3 ) )
= ( ( inj_on6435365835345961783et_int @ F @ A3 )
& ~ ( member_set_int @ ( F @ A ) @ ( image_524474410958335435et_int @ F @ ( minus_8897228262479074673et_int @ A3 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_956_inj__on__insert,axiom,
! [F: set_int > int > set_int,A: set_int,A3: set_set_int] :
( ( inj_on5988255161265977986et_int @ F @ ( insert_set_int @ A @ A3 ) )
= ( ( inj_on5988255161265977986et_int @ F @ A3 )
& ~ ( member_int_set_int @ ( F @ A ) @ ( image_1787253287920604950et_int @ F @ ( minus_8897228262479074673et_int @ A3 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_957_inj__on__insert,axiom,
! [F: set_int > set_set_int,A: set_int,A3: set_set_int] :
( ( inj_on6285404204842837485et_int @ F @ ( insert_set_int @ A @ A3 ) )
= ( ( inj_on6285404204842837485et_int @ F @ A3 )
& ~ ( member_set_set_int @ ( F @ A ) @ ( image_1010086626112315521et_int @ F @ ( minus_8897228262479074673et_int @ A3 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_958_inj__on__insert,axiom,
! [F: set_int > nat,A: set_int,A3: set_set_int] :
( ( inj_on_set_int_nat @ F @ ( insert_set_int @ A @ A3 ) )
= ( ( inj_on_set_int_nat @ F @ A3 )
& ~ ( member_nat @ ( F @ A ) @ ( image_set_int_nat @ F @ ( minus_8897228262479074673et_int @ A3 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_959_inj__on__empty,axiom,
! [F: int > set_int] : ( inj_on_int_set_int @ F @ bot_bot_set_int ) ).
% inj_on_empty
thf(fact_960_inj__on__empty,axiom,
! [F: int > int] : ( inj_on_int_int @ F @ bot_bot_set_int ) ).
% inj_on_empty
thf(fact_961_inj__on__empty,axiom,
! [F: set_int > set_int] : ( inj_on6435365835345961783et_int @ F @ bot_bot_set_set_int ) ).
% inj_on_empty
thf(fact_962_inj__uminus,axiom,
! [A3: set_int] : ( inj_on_int_int @ uminus_uminus_int @ A3 ) ).
% inj_uminus
thf(fact_963_surj__uminus,axiom,
( ( image_int_int @ uminus_uminus_int @ top_top_set_int )
= top_top_set_int ) ).
% surj_uminus
thf(fact_964_diff__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_965_inj__on__diff,axiom,
! [F: int > set_int,A3: set_int,B2: set_int] :
( ( inj_on_int_set_int @ F @ A3 )
=> ( inj_on_int_set_int @ F @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ).
% inj_on_diff
thf(fact_966_inj__on__diff,axiom,
! [F: set_int > set_int,A3: set_set_int,B2: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ A3 )
=> ( inj_on6435365835345961783et_int @ F @ ( minus_8897228262479074673et_int @ A3 @ B2 ) ) ) ).
% inj_on_diff
thf(fact_967_inj__on__diff,axiom,
! [F: int > int,A3: set_int,B2: set_int] :
( ( inj_on_int_int @ F @ A3 )
=> ( inj_on_int_int @ F @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ).
% inj_on_diff
thf(fact_968_surj__def,axiom,
! [F: set_int > set_set_int] :
( ( ( image_1010086626112315521et_int @ F @ top_top_set_set_int )
= top_to5524576366173240574et_int )
= ( ! [Y3: set_set_int] :
? [X5: set_int] :
( Y3
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_969_surj__def,axiom,
! [F: int > set_int] :
( ( ( image_int_set_int @ F @ top_top_set_int )
= top_top_set_set_int )
= ( ! [Y3: set_int] :
? [X5: int] :
( Y3
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_970_surj__def,axiom,
! [F: int > int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
= ( ! [Y3: int] :
? [X5: int] :
( Y3
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_971_surj__def,axiom,
! [F: int > nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X5: int] :
( Y3
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_972_surj__def,axiom,
! [F: nat > int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
= ( ! [Y3: int] :
? [X5: nat] :
( Y3
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_973_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X5: nat] :
( Y3
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_974_surjI,axiom,
! [G: set_int > set_set_int,F: set_set_int > set_int] :
( ! [X4: set_set_int] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_1010086626112315521et_int @ G @ top_top_set_set_int )
= top_to5524576366173240574et_int ) ) ).
% surjI
thf(fact_975_surjI,axiom,
! [G: int > set_int,F: set_int > int] :
( ! [X4: set_int] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_int_set_int @ G @ top_top_set_int )
= top_top_set_set_int ) ) ).
% surjI
thf(fact_976_surjI,axiom,
! [G: int > int,F: int > int] :
( ! [X4: int] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_int_int @ G @ top_top_set_int )
= top_top_set_int ) ) ).
% surjI
thf(fact_977_surjI,axiom,
! [G: int > nat,F: nat > int] :
( ! [X4: nat] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_int_nat @ G @ top_top_set_int )
= top_top_set_nat ) ) ).
% surjI
thf(fact_978_surjI,axiom,
! [G: nat > int,F: int > nat] :
( ! [X4: int] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_nat_int @ G @ top_top_set_nat )
= top_top_set_int ) ) ).
% surjI
thf(fact_979_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X4: nat] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_980_surjE,axiom,
! [F: set_int > set_set_int,Y: set_set_int] :
( ( ( image_1010086626112315521et_int @ F @ top_top_set_set_int )
= top_to5524576366173240574et_int )
=> ~ ! [X4: set_int] :
( Y
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_981_surjE,axiom,
! [F: int > set_int,Y: set_int] :
( ( ( image_int_set_int @ F @ top_top_set_int )
= top_top_set_set_int )
=> ~ ! [X4: int] :
( Y
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_982_surjE,axiom,
! [F: int > int,Y: int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ~ ! [X4: int] :
( Y
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_983_surjE,axiom,
! [F: int > nat,Y: nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
=> ~ ! [X4: int] :
( Y
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_984_surjE,axiom,
! [F: nat > int,Y: int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ~ ! [X4: nat] :
( Y
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_985_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X4: nat] :
( Y
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_986_surjD,axiom,
! [F: set_int > set_set_int,Y: set_set_int] :
( ( ( image_1010086626112315521et_int @ F @ top_top_set_set_int )
= top_to5524576366173240574et_int )
=> ? [X4: set_int] :
( Y
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_987_surjD,axiom,
! [F: int > set_int,Y: set_int] :
( ( ( image_int_set_int @ F @ top_top_set_int )
= top_top_set_set_int )
=> ? [X4: int] :
( Y
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_988_surjD,axiom,
! [F: int > int,Y: int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ? [X4: int] :
( Y
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_989_surjD,axiom,
! [F: int > nat,Y: nat] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
=> ? [X4: int] :
( Y
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_990_surjD,axiom,
! [F: nat > int,Y: int] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ? [X4: nat] :
( Y
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_991_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X4: nat] :
( Y
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_992_translation__diff,axiom,
! [A: int,S2: set_int,T: set_int] :
( ( image_int_int @ ( plus_plus_int @ A ) @ ( minus_minus_set_int @ S2 @ T ) )
= ( minus_minus_set_int @ ( image_int_int @ ( plus_plus_int @ A ) @ S2 ) @ ( image_int_int @ ( plus_plus_int @ A ) @ T ) ) ) ).
% translation_diff
thf(fact_993_translation__Compl,axiom,
! [A: int,T: set_int] :
( ( image_int_int @ ( plus_plus_int @ A ) @ ( uminus1532241313380277803et_int @ T ) )
= ( uminus1532241313380277803et_int @ ( image_int_int @ ( plus_plus_int @ A ) @ T ) ) ) ).
% translation_Compl
thf(fact_994_inj__def,axiom,
! [F: set_int > set_int] :
( ( inj_on6435365835345961783et_int @ F @ top_top_set_set_int )
= ( ! [X5: set_int,Y3: set_int] :
( ( ( F @ X5 )
= ( F @ Y3 ) )
=> ( X5 = Y3 ) ) ) ) ).
% inj_def
thf(fact_995_inj__def,axiom,
! [F: int > set_int] :
( ( inj_on_int_set_int @ F @ top_top_set_int )
= ( ! [X5: int,Y3: int] :
( ( ( F @ X5 )
= ( F @ Y3 ) )
=> ( X5 = Y3 ) ) ) ) ).
% inj_def
thf(fact_996_inj__def,axiom,
! [F: int > int] :
( ( inj_on_int_int @ F @ top_top_set_int )
= ( ! [X5: int,Y3: int] :
( ( ( F @ X5 )
= ( F @ Y3 ) )
=> ( X5 = Y3 ) ) ) ) ).
% inj_def
thf(fact_997_inj__eq,axiom,
! [F: set_int > set_int,X: set_int,Y: set_int] :
( ( inj_on6435365835345961783et_int @ F @ top_top_set_set_int )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_998_inj__eq,axiom,
! [F: int > set_int,X: int,Y: int] :
( ( inj_on_int_set_int @ F @ top_top_set_int )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_999_inj__eq,axiom,
! [F: int > int,X: int,Y: int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_1000_injI,axiom,
! [F: set_int > set_int] :
( ! [X4: set_int,Y4: set_int] :
( ( ( F @ X4 )
= ( F @ Y4 ) )
=> ( X4 = Y4 ) )
=> ( inj_on6435365835345961783et_int @ F @ top_top_set_set_int ) ) ).
% injI
thf(fact_1001_injI,axiom,
! [F: int > set_int] :
( ! [X4: int,Y4: int] :
( ( ( F @ X4 )
= ( F @ Y4 ) )
=> ( X4 = Y4 ) )
=> ( inj_on_int_set_int @ F @ top_top_set_int ) ) ).
% injI
thf(fact_1002_injI,axiom,
! [F: int > int] :
( ! [X4: int,Y4: int] :
( ( ( F @ X4 )
= ( F @ Y4 ) )
=> ( X4 = Y4 ) )
=> ( inj_on_int_int @ F @ top_top_set_int ) ) ).
% injI
thf(fact_1003_injD,axiom,
! [F: set_int > set_int,X: set_int,Y: set_int] :
( ( inj_on6435365835345961783et_int @ F @ top_top_set_set_int )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_1004_injD,axiom,
! [F: int > set_int,X: int,Y: int] :
( ( inj_on_int_set_int @ F @ top_top_set_int )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_1005_injD,axiom,
! [F: int > int,X: int,Y: int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_1006_inj__on__add,axiom,
! [A: int,A3: set_int] : ( inj_on_int_int @ ( plus_plus_int @ A ) @ A3 ) ).
% inj_on_add
thf(fact_1007_inj__on__add,axiom,
! [A: nat,A3: set_nat] : ( inj_on_nat_nat @ ( plus_plus_nat @ A ) @ A3 ) ).
% inj_on_add
thf(fact_1008_linorder__inj__onI_H,axiom,
! [A3: set_int,F: int > set_int] :
( ! [I3: int,J2: int] :
( ( member_int @ I3 @ A3 )
=> ( ( member_int @ J2 @ A3 )
=> ( ( ord_less_int @ I3 @ J2 )
=> ( ( F @ I3 )
!= ( F @ J2 ) ) ) ) )
=> ( inj_on_int_set_int @ F @ A3 ) ) ).
% linorder_inj_onI'
thf(fact_1009_linorder__inj__onI_H,axiom,
! [A3: set_int,F: int > int] :
( ! [I3: int,J2: int] :
( ( member_int @ I3 @ A3 )
=> ( ( member_int @ J2 @ A3 )
=> ( ( ord_less_int @ I3 @ J2 )
=> ( ( F @ I3 )
!= ( F @ J2 ) ) ) ) )
=> ( inj_on_int_int @ F @ A3 ) ) ).
% linorder_inj_onI'
thf(fact_1010_subset__inj__on,axiom,
! [F: int > set_int,B2: set_int,A3: set_int] :
( ( inj_on_int_set_int @ F @ B2 )
=> ( ( ord_less_eq_set_int @ A3 @ B2 )
=> ( inj_on_int_set_int @ F @ A3 ) ) ) ).
% subset_inj_on
thf(fact_1011_subset__inj__on,axiom,
! [F: int > int,B2: set_int,A3: set_int] :
( ( inj_on_int_int @ F @ B2 )
=> ( ( ord_less_eq_set_int @ A3 @ B2 )
=> ( inj_on_int_int @ F @ A3 ) ) ) ).
% subset_inj_on
thf(fact_1012_subset__inj__on,axiom,
! [F: set_int > set_int,B2: set_set_int,A3: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ B2 )
=> ( ( ord_le4403425263959731960et_int @ A3 @ B2 )
=> ( inj_on6435365835345961783et_int @ F @ A3 ) ) ) ).
% subset_inj_on
thf(fact_1013_inj__on__subset,axiom,
! [F: int > set_int,A3: set_int,B2: set_int] :
( ( inj_on_int_set_int @ F @ A3 )
=> ( ( ord_less_eq_set_int @ B2 @ A3 )
=> ( inj_on_int_set_int @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_1014_inj__on__subset,axiom,
! [F: int > int,A3: set_int,B2: set_int] :
( ( inj_on_int_int @ F @ A3 )
=> ( ( ord_less_eq_set_int @ B2 @ A3 )
=> ( inj_on_int_int @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_1015_inj__on__subset,axiom,
! [F: set_int > set_int,A3: set_set_int,B2: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ A3 )
=> ( ( ord_le4403425263959731960et_int @ B2 @ A3 )
=> ( inj_on6435365835345961783et_int @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_1016_inj__on__diff__left,axiom,
! [A: int,A3: set_int] : ( inj_on_int_int @ ( minus_minus_int @ A ) @ A3 ) ).
% inj_on_diff_left
thf(fact_1017_inj__on__image__iff,axiom,
! [A3: set_set_int,G: set_int > set_int,F: set_int > set_int] :
( ! [X4: set_int] :
( ( member_set_int @ X4 @ A3 )
=> ! [Xa: set_int] :
( ( member_set_int @ Xa @ A3 )
=> ( ( ( G @ ( F @ X4 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X4 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on6435365835345961783et_int @ F @ A3 )
=> ( ( inj_on6435365835345961783et_int @ G @ ( image_524474410958335435et_int @ F @ A3 ) )
= ( inj_on6435365835345961783et_int @ G @ A3 ) ) ) ) ).
% inj_on_image_iff
thf(fact_1018_inj__on__image__iff,axiom,
! [A3: set_int,G: int > set_int,F: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A3 )
=> ! [Xa: int] :
( ( member_int @ Xa @ A3 )
=> ( ( ( G @ ( F @ X4 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X4 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_int_int @ F @ A3 )
=> ( ( inj_on_int_set_int @ G @ ( image_int_int @ F @ A3 ) )
= ( inj_on_int_set_int @ G @ A3 ) ) ) ) ).
% inj_on_image_iff
thf(fact_1019_inj__on__image__iff,axiom,
! [A3: set_int,G: int > int,F: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ A3 )
=> ! [Xa: int] :
( ( member_int @ Xa @ A3 )
=> ( ( ( G @ ( F @ X4 ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X4 )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on_int_int @ F @ A3 )
=> ( ( inj_on_int_int @ G @ ( image_int_int @ F @ A3 ) )
= ( inj_on_int_int @ G @ A3 ) ) ) ) ).
% inj_on_image_iff
thf(fact_1020_image__strict__mono,axiom,
! [F: set_int > set_set_int,B2: set_set_int,A3: set_set_int] :
( ( inj_on6285404204842837485et_int @ F @ B2 )
=> ( ( ord_less_set_set_int @ A3 @ B2 )
=> ( ord_le4562804192517611682et_int @ ( image_1010086626112315521et_int @ F @ A3 ) @ ( image_1010086626112315521et_int @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_1021_image__strict__mono,axiom,
! [F: int > set_int,B2: set_int,A3: set_int] :
( ( inj_on_int_set_int @ F @ B2 )
=> ( ( ord_less_set_int @ A3 @ B2 )
=> ( ord_less_set_set_int @ ( image_int_set_int @ F @ A3 ) @ ( image_int_set_int @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_1022_image__strict__mono,axiom,
! [F: set_int > set_int,B2: set_set_int,A3: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ B2 )
=> ( ( ord_less_set_set_int @ A3 @ B2 )
=> ( ord_less_set_set_int @ ( image_524474410958335435et_int @ F @ A3 ) @ ( image_524474410958335435et_int @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_1023_image__strict__mono,axiom,
! [F: int > int,B2: set_int,A3: set_int] :
( ( inj_on_int_int @ F @ B2 )
=> ( ( ord_less_set_int @ A3 @ B2 )
=> ( ord_less_set_int @ ( image_int_int @ F @ A3 ) @ ( image_int_int @ F @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_1024_surj__Compl__image__subset,axiom,
! [F: int > nat,A3: set_int] :
( ( ( image_int_nat @ F @ top_top_set_int )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_int_nat @ F @ A3 ) ) @ ( image_int_nat @ F @ ( uminus1532241313380277803et_int @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1025_surj__Compl__image__subset,axiom,
! [F: nat > nat,A3: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ ( image_nat_nat @ F @ A3 ) ) @ ( image_nat_nat @ F @ ( uminus5710092332889474511et_nat @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1026_surj__Compl__image__subset,axiom,
! [F: int > int,A3: set_int] :
( ( ( image_int_int @ F @ top_top_set_int )
= top_top_set_int )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ ( image_int_int @ F @ A3 ) ) @ ( image_int_int @ F @ ( uminus1532241313380277803et_int @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1027_surj__Compl__image__subset,axiom,
! [F: nat > int,A3: set_nat] :
( ( ( image_nat_int @ F @ top_top_set_nat )
= top_top_set_int )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ ( image_nat_int @ F @ A3 ) ) @ ( image_nat_int @ F @ ( uminus5710092332889474511et_nat @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1028_surj__Compl__image__subset,axiom,
! [F: int > set_int,A3: set_int] :
( ( ( image_int_set_int @ F @ top_top_set_int )
= top_top_set_set_int )
=> ( ord_le4403425263959731960et_int @ ( uminus7346710233107665121et_int @ ( image_int_set_int @ F @ A3 ) ) @ ( image_int_set_int @ F @ ( uminus1532241313380277803et_int @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1029_surj__Compl__image__subset,axiom,
! [F: nat > set_int,A3: set_nat] :
( ( ( image_nat_set_int @ F @ top_top_set_nat )
= top_top_set_set_int )
=> ( ord_le4403425263959731960et_int @ ( uminus7346710233107665121et_int @ ( image_nat_set_int @ F @ A3 ) ) @ ( image_nat_set_int @ F @ ( uminus5710092332889474511et_nat @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1030_surj__Compl__image__subset,axiom,
! [F: set_int > set_set_int,A3: set_set_int] :
( ( ( image_1010086626112315521et_int @ F @ top_top_set_set_int )
= top_to5524576366173240574et_int )
=> ( ord_le4317611570275147438et_int @ ( uminus3047225438891465879et_int @ ( image_1010086626112315521et_int @ F @ A3 ) ) @ ( image_1010086626112315521et_int @ F @ ( uminus7346710233107665121et_int @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1031_surj__Compl__image__subset,axiom,
! [F: int > set_set_int,A3: set_int] :
( ( ( image_6617583118289273547et_int @ F @ top_top_set_int )
= top_to5524576366173240574et_int )
=> ( ord_le4317611570275147438et_int @ ( uminus3047225438891465879et_int @ ( image_6617583118289273547et_int @ F @ A3 ) ) @ ( image_6617583118289273547et_int @ F @ ( uminus1532241313380277803et_int @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1032_surj__Compl__image__subset,axiom,
! [F: nat > set_set_int,A3: set_nat] :
( ( ( image_8927401050382224495et_int @ F @ top_top_set_nat )
= top_to5524576366173240574et_int )
=> ( ord_le4317611570275147438et_int @ ( uminus3047225438891465879et_int @ ( image_8927401050382224495et_int @ F @ A3 ) ) @ ( image_8927401050382224495et_int @ F @ ( uminus5710092332889474511et_nat @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1033_linorder__inj__onI,axiom,
! [A3: set_set_int,F: set_int > set_int] :
( ! [X4: set_int,Y4: set_int] :
( ( ord_less_set_int @ X4 @ Y4 )
=> ( ( member_set_int @ X4 @ A3 )
=> ( ( member_set_int @ Y4 @ A3 )
=> ( ( F @ X4 )
!= ( F @ Y4 ) ) ) ) )
=> ( ! [X4: set_int,Y4: set_int] :
( ( member_set_int @ X4 @ A3 )
=> ( ( member_set_int @ Y4 @ A3 )
=> ( ( ord_less_eq_set_int @ X4 @ Y4 )
| ( ord_less_eq_set_int @ Y4 @ X4 ) ) ) )
=> ( inj_on6435365835345961783et_int @ F @ A3 ) ) ) ).
% linorder_inj_onI
thf(fact_1034_linorder__inj__onI,axiom,
! [A3: set_int,F: int > set_int] :
( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ( member_int @ X4 @ A3 )
=> ( ( member_int @ Y4 @ A3 )
=> ( ( F @ X4 )
!= ( F @ Y4 ) ) ) ) )
=> ( ! [X4: int,Y4: int] :
( ( member_int @ X4 @ A3 )
=> ( ( member_int @ Y4 @ A3 )
=> ( ( ord_less_eq_int @ X4 @ Y4 )
| ( ord_less_eq_int @ Y4 @ X4 ) ) ) )
=> ( inj_on_int_set_int @ F @ A3 ) ) ) ).
% linorder_inj_onI
thf(fact_1035_linorder__inj__onI,axiom,
! [A3: set_int,F: int > int] :
( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ( member_int @ X4 @ A3 )
=> ( ( member_int @ Y4 @ A3 )
=> ( ( F @ X4 )
!= ( F @ Y4 ) ) ) ) )
=> ( ! [X4: int,Y4: int] :
( ( member_int @ X4 @ A3 )
=> ( ( member_int @ Y4 @ A3 )
=> ( ( ord_less_eq_int @ X4 @ Y4 )
| ( ord_less_eq_int @ Y4 @ X4 ) ) ) )
=> ( inj_on_int_int @ F @ A3 ) ) ) ).
% linorder_inj_onI
thf(fact_1036_linorder__injI,axiom,
! [F: int > set_int] :
( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ( F @ X4 )
!= ( F @ Y4 ) ) )
=> ( inj_on_int_set_int @ F @ top_top_set_int ) ) ).
% linorder_injI
thf(fact_1037_linorder__injI,axiom,
! [F: int > int] :
( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ( F @ X4 )
!= ( F @ Y4 ) ) )
=> ( inj_on_int_int @ F @ top_top_set_int ) ) ).
% linorder_injI
thf(fact_1038_inj__image__mem__iff,axiom,
! [F: int > int,A: int,A3: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( member_int @ ( F @ A ) @ ( image_int_int @ F @ A3 ) )
= ( member_int @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_1039_inj__image__mem__iff,axiom,
! [F: int > nat,A: int,A3: set_int] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( member_nat @ ( F @ A ) @ ( image_int_nat @ F @ A3 ) )
= ( member_int @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_1040_inj__image__mem__iff,axiom,
! [F: nat > int,A: nat,A3: set_nat] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( member_int @ ( F @ A ) @ ( image_nat_int @ F @ A3 ) )
= ( member_nat @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_1041_inj__image__mem__iff,axiom,
! [F: nat > nat,A: nat,A3: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ ( F @ A ) @ ( image_nat_nat @ F @ A3 ) )
= ( member_nat @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_1042_inj__image__mem__iff,axiom,
! [F: set_int > int,A: set_int,A3: set_set_int] :
( ( inj_on_set_int_int @ F @ top_top_set_set_int )
=> ( ( member_int @ ( F @ A ) @ ( image_set_int_int @ F @ A3 ) )
= ( member_set_int @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_1043_inj__image__mem__iff,axiom,
! [F: set_int > nat,A: set_int,A3: set_set_int] :
( ( inj_on_set_int_nat @ F @ top_top_set_set_int )
=> ( ( member_nat @ ( F @ A ) @ ( image_set_int_nat @ F @ A3 ) )
= ( member_set_int @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_1044_inj__image__mem__iff,axiom,
! [F: int > set_int,A: int,A3: set_int] :
( ( inj_on_int_set_int @ F @ top_top_set_int )
=> ( ( member_set_int @ ( F @ A ) @ ( image_int_set_int @ F @ A3 ) )
= ( member_int @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_1045_inj__image__mem__iff,axiom,
! [F: nat > set_int,A: nat,A3: set_nat] :
( ( inj_on_nat_set_int @ F @ top_top_set_nat )
=> ( ( member_set_int @ ( F @ A ) @ ( image_nat_set_int @ F @ A3 ) )
= ( member_nat @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_1046_inj__image__mem__iff,axiom,
! [F: set_set_int > int,A: set_set_int,A3: set_set_set_int] :
( ( inj_on1358648829685248055nt_int @ F @ top_to5524576366173240574et_int )
=> ( ( member_int @ ( F @ A ) @ ( image_4671129442152397515nt_int @ F @ A3 ) )
= ( member_set_set_int @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_1047_inj__image__mem__iff,axiom,
! [F: set_int > set_int,A: set_int,A3: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ top_top_set_set_int )
=> ( ( member_set_int @ ( F @ A ) @ ( image_524474410958335435et_int @ F @ A3 ) )
= ( member_set_int @ A @ A3 ) ) ) ).
% inj_image_mem_iff
thf(fact_1048_inj__image__eq__iff,axiom,
! [F: set_int > set_set_int,A3: set_set_int,B2: set_set_int] :
( ( inj_on6285404204842837485et_int @ F @ top_top_set_set_int )
=> ( ( ( image_1010086626112315521et_int @ F @ A3 )
= ( image_1010086626112315521et_int @ F @ B2 ) )
= ( A3 = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_1049_inj__image__eq__iff,axiom,
! [F: set_int > set_int,A3: set_set_int,B2: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ top_top_set_set_int )
=> ( ( ( image_524474410958335435et_int @ F @ A3 )
= ( image_524474410958335435et_int @ F @ B2 ) )
= ( A3 = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_1050_inj__image__eq__iff,axiom,
! [F: int > set_int,A3: set_int,B2: set_int] :
( ( inj_on_int_set_int @ F @ top_top_set_int )
=> ( ( ( image_int_set_int @ F @ A3 )
= ( image_int_set_int @ F @ B2 ) )
= ( A3 = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_1051_inj__image__eq__iff,axiom,
! [F: int > int,A3: set_int,B2: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( ( image_int_int @ F @ A3 )
= ( image_int_int @ F @ B2 ) )
= ( A3 = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_1052_range__ex1__eq,axiom,
! [F: int > int,B: int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( member_int @ B @ ( image_int_int @ F @ top_top_set_int ) )
= ( ? [X5: int] :
( ( B
= ( F @ X5 ) )
& ! [Y3: int] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1053_range__ex1__eq,axiom,
! [F: int > nat,B: nat] :
( ( inj_on_int_nat @ F @ top_top_set_int )
=> ( ( member_nat @ B @ ( image_int_nat @ F @ top_top_set_int ) )
= ( ? [X5: int] :
( ( B
= ( F @ X5 ) )
& ! [Y3: int] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1054_range__ex1__eq,axiom,
! [F: nat > int,B: int] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( member_int @ B @ ( image_nat_int @ F @ top_top_set_nat ) )
= ( ? [X5: nat] :
( ( B
= ( F @ X5 ) )
& ! [Y3: nat] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1055_range__ex1__eq,axiom,
! [F: nat > nat,B: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) )
= ( ? [X5: nat] :
( ( B
= ( F @ X5 ) )
& ! [Y3: nat] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1056_range__ex1__eq,axiom,
! [F: int > set_int,B: set_int] :
( ( inj_on_int_set_int @ F @ top_top_set_int )
=> ( ( member_set_int @ B @ ( image_int_set_int @ F @ top_top_set_int ) )
= ( ? [X5: int] :
( ( B
= ( F @ X5 ) )
& ! [Y3: int] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1057_range__ex1__eq,axiom,
! [F: nat > set_int,B: set_int] :
( ( inj_on_nat_set_int @ F @ top_top_set_nat )
=> ( ( member_set_int @ B @ ( image_nat_set_int @ F @ top_top_set_nat ) )
= ( ? [X5: nat] :
( ( B
= ( F @ X5 ) )
& ! [Y3: nat] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1058_range__ex1__eq,axiom,
! [F: set_int > set_int,B: set_int] :
( ( inj_on6435365835345961783et_int @ F @ top_top_set_set_int )
=> ( ( member_set_int @ B @ ( image_524474410958335435et_int @ F @ top_top_set_set_int ) )
= ( ? [X5: set_int] :
( ( B
= ( F @ X5 ) )
& ! [Y3: set_int] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1059_range__ex1__eq,axiom,
! [F: int > set_set_int,B: set_set_int] :
( ( inj_on3305102505822124087et_int @ F @ top_top_set_int )
=> ( ( member_set_set_int @ B @ ( image_6617583118289273547et_int @ F @ top_top_set_int ) )
= ( ? [X5: int] :
( ( B
= ( F @ X5 ) )
& ! [Y3: int] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1060_range__ex1__eq,axiom,
! [F: nat > set_set_int,B: set_set_int] :
( ( inj_on5614920437915075035et_int @ F @ top_top_set_nat )
=> ( ( member_set_set_int @ B @ ( image_8927401050382224495et_int @ F @ top_top_set_nat ) )
= ( ? [X5: nat] :
( ( B
= ( F @ X5 ) )
& ! [Y3: nat] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1061_range__ex1__eq,axiom,
! [F: set_int > set_set_int,B: set_set_int] :
( ( inj_on6285404204842837485et_int @ F @ top_top_set_set_int )
=> ( ( member_set_set_int @ B @ ( image_1010086626112315521et_int @ F @ top_top_set_set_int ) )
= ( ? [X5: set_int] :
( ( B
= ( F @ X5 ) )
& ! [Y3: set_int] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X5 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1062_image__set__diff,axiom,
! [F: set_int > set_set_int,A3: set_set_int,B2: set_set_int] :
( ( inj_on6285404204842837485et_int @ F @ top_top_set_set_int )
=> ( ( image_1010086626112315521et_int @ F @ ( minus_8897228262479074673et_int @ A3 @ B2 ) )
= ( minus_6857623457997529383et_int @ ( image_1010086626112315521et_int @ F @ A3 ) @ ( image_1010086626112315521et_int @ F @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_1063_image__set__diff,axiom,
! [F: set_int > set_int,A3: set_set_int,B2: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ top_top_set_set_int )
=> ( ( image_524474410958335435et_int @ F @ ( minus_8897228262479074673et_int @ A3 @ B2 ) )
= ( minus_8897228262479074673et_int @ ( image_524474410958335435et_int @ F @ A3 ) @ ( image_524474410958335435et_int @ F @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_1064_image__set__diff,axiom,
! [F: int > set_int,A3: set_int,B2: set_int] :
( ( inj_on_int_set_int @ F @ top_top_set_int )
=> ( ( image_int_set_int @ F @ ( minus_minus_set_int @ A3 @ B2 ) )
= ( minus_8897228262479074673et_int @ ( image_int_set_int @ F @ A3 ) @ ( image_int_set_int @ F @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_1065_image__set__diff,axiom,
! [F: int > int,A3: set_int,B2: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( image_int_int @ F @ ( minus_minus_set_int @ A3 @ B2 ) )
= ( minus_minus_set_int @ ( image_int_int @ F @ A3 ) @ ( image_int_int @ F @ B2 ) ) ) ) ).
% image_set_diff
thf(fact_1066_inj__on__image__eq__iff,axiom,
! [F: int > set_int,C2: set_int,A3: set_int,B2: set_int] :
( ( inj_on_int_set_int @ F @ C2 )
=> ( ( ord_less_eq_set_int @ A3 @ C2 )
=> ( ( ord_less_eq_set_int @ B2 @ C2 )
=> ( ( ( image_int_set_int @ F @ A3 )
= ( image_int_set_int @ F @ B2 ) )
= ( A3 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1067_inj__on__image__eq__iff,axiom,
! [F: int > int,C2: set_int,A3: set_int,B2: set_int] :
( ( inj_on_int_int @ F @ C2 )
=> ( ( ord_less_eq_set_int @ A3 @ C2 )
=> ( ( ord_less_eq_set_int @ B2 @ C2 )
=> ( ( ( image_int_int @ F @ A3 )
= ( image_int_int @ F @ B2 ) )
= ( A3 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1068_inj__on__image__eq__iff,axiom,
! [F: set_int > set_set_int,C2: set_set_int,A3: set_set_int,B2: set_set_int] :
( ( inj_on6285404204842837485et_int @ F @ C2 )
=> ( ( ord_le4403425263959731960et_int @ A3 @ C2 )
=> ( ( ord_le4403425263959731960et_int @ B2 @ C2 )
=> ( ( ( image_1010086626112315521et_int @ F @ A3 )
= ( image_1010086626112315521et_int @ F @ B2 ) )
= ( A3 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1069_inj__on__image__eq__iff,axiom,
! [F: set_int > set_int,C2: set_set_int,A3: set_set_int,B2: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ C2 )
=> ( ( ord_le4403425263959731960et_int @ A3 @ C2 )
=> ( ( ord_le4403425263959731960et_int @ B2 @ C2 )
=> ( ( ( image_524474410958335435et_int @ F @ A3 )
= ( image_524474410958335435et_int @ F @ B2 ) )
= ( A3 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1070_inj__on__image__mem__iff,axiom,
! [F: nat > int,B2: set_nat,A: nat,A3: set_nat] :
( ( inj_on_nat_int @ F @ B2 )
=> ( ( member_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( member_int @ ( F @ A ) @ ( image_nat_int @ F @ A3 ) )
= ( member_nat @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1071_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B2: set_nat,A: nat,A3: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( member_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( member_nat @ ( F @ A ) @ ( image_nat_nat @ F @ A3 ) )
= ( member_nat @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1072_inj__on__image__mem__iff,axiom,
! [F: int > int,B2: set_int,A: int,A3: set_int] :
( ( inj_on_int_int @ F @ B2 )
=> ( ( member_int @ A @ B2 )
=> ( ( ord_less_eq_set_int @ A3 @ B2 )
=> ( ( member_int @ ( F @ A ) @ ( image_int_int @ F @ A3 ) )
= ( member_int @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1073_inj__on__image__mem__iff,axiom,
! [F: int > nat,B2: set_int,A: int,A3: set_int] :
( ( inj_on_int_nat @ F @ B2 )
=> ( ( member_int @ A @ B2 )
=> ( ( ord_less_eq_set_int @ A3 @ B2 )
=> ( ( member_nat @ ( F @ A ) @ ( image_int_nat @ F @ A3 ) )
= ( member_int @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1074_inj__on__image__mem__iff,axiom,
! [F: nat > set_int,B2: set_nat,A: nat,A3: set_nat] :
( ( inj_on_nat_set_int @ F @ B2 )
=> ( ( member_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( member_set_int @ ( F @ A ) @ ( image_nat_set_int @ F @ A3 ) )
= ( member_nat @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1075_inj__on__image__mem__iff,axiom,
! [F: int > set_int,B2: set_int,A: int,A3: set_int] :
( ( inj_on_int_set_int @ F @ B2 )
=> ( ( member_int @ A @ B2 )
=> ( ( ord_less_eq_set_int @ A3 @ B2 )
=> ( ( member_set_int @ ( F @ A ) @ ( image_int_set_int @ F @ A3 ) )
= ( member_int @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1076_inj__on__image__mem__iff,axiom,
! [F: set_int > int,B2: set_set_int,A: set_int,A3: set_set_int] :
( ( inj_on_set_int_int @ F @ B2 )
=> ( ( member_set_int @ A @ B2 )
=> ( ( ord_le4403425263959731960et_int @ A3 @ B2 )
=> ( ( member_int @ ( F @ A ) @ ( image_set_int_int @ F @ A3 ) )
= ( member_set_int @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1077_inj__on__image__mem__iff,axiom,
! [F: set_int > nat,B2: set_set_int,A: set_int,A3: set_set_int] :
( ( inj_on_set_int_nat @ F @ B2 )
=> ( ( member_set_int @ A @ B2 )
=> ( ( ord_le4403425263959731960et_int @ A3 @ B2 )
=> ( ( member_nat @ ( F @ A ) @ ( image_set_int_nat @ F @ A3 ) )
= ( member_set_int @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1078_inj__on__image__mem__iff,axiom,
! [F: nat > set_set_int,B2: set_nat,A: nat,A3: set_nat] :
( ( inj_on5614920437915075035et_int @ F @ B2 )
=> ( ( member_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( member_set_set_int @ ( F @ A ) @ ( image_8927401050382224495et_int @ F @ A3 ) )
= ( member_nat @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1079_inj__on__image__mem__iff,axiom,
! [F: int > set_set_int,B2: set_int,A: int,A3: set_int] :
( ( inj_on3305102505822124087et_int @ F @ B2 )
=> ( ( member_int @ A @ B2 )
=> ( ( ord_less_eq_set_int @ A3 @ B2 )
=> ( ( member_set_set_int @ ( F @ A ) @ ( image_6617583118289273547et_int @ F @ A3 ) )
= ( member_int @ A @ A3 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1080_inj__on__image__set__diff,axiom,
! [F: int > set_int,C2: set_int,A3: set_int,B2: set_int] :
( ( inj_on_int_set_int @ F @ C2 )
=> ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B2 ) @ C2 )
=> ( ( ord_less_eq_set_int @ B2 @ C2 )
=> ( ( image_int_set_int @ F @ ( minus_minus_set_int @ A3 @ B2 ) )
= ( minus_8897228262479074673et_int @ ( image_int_set_int @ F @ A3 ) @ ( image_int_set_int @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1081_inj__on__image__set__diff,axiom,
! [F: int > int,C2: set_int,A3: set_int,B2: set_int] :
( ( inj_on_int_int @ F @ C2 )
=> ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B2 ) @ C2 )
=> ( ( ord_less_eq_set_int @ B2 @ C2 )
=> ( ( image_int_int @ F @ ( minus_minus_set_int @ A3 @ B2 ) )
= ( minus_minus_set_int @ ( image_int_int @ F @ A3 ) @ ( image_int_int @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1082_inj__on__image__set__diff,axiom,
! [F: set_int > set_set_int,C2: set_set_int,A3: set_set_int,B2: set_set_int] :
( ( inj_on6285404204842837485et_int @ F @ C2 )
=> ( ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A3 @ B2 ) @ C2 )
=> ( ( ord_le4403425263959731960et_int @ B2 @ C2 )
=> ( ( image_1010086626112315521et_int @ F @ ( minus_8897228262479074673et_int @ A3 @ B2 ) )
= ( minus_6857623457997529383et_int @ ( image_1010086626112315521et_int @ F @ A3 ) @ ( image_1010086626112315521et_int @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1083_inj__on__image__set__diff,axiom,
! [F: set_int > set_int,C2: set_set_int,A3: set_set_int,B2: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ C2 )
=> ( ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A3 @ B2 ) @ C2 )
=> ( ( ord_le4403425263959731960et_int @ B2 @ C2 )
=> ( ( image_524474410958335435et_int @ F @ ( minus_8897228262479074673et_int @ A3 @ B2 ) )
= ( minus_8897228262479074673et_int @ ( image_524474410958335435et_int @ F @ A3 ) @ ( image_524474410958335435et_int @ F @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1084_inj__img__insertE,axiom,
! [F: int > int,A3: set_int,X: int,B2: set_int] :
( ( inj_on_int_int @ F @ A3 )
=> ( ~ ( member_int @ X @ B2 )
=> ( ( ( insert_int @ X @ B2 )
= ( image_int_int @ F @ A3 ) )
=> ~ ! [X9: int,A6: set_int] :
( ~ ( member_int @ X9 @ A6 )
=> ( ( A3
= ( insert_int @ X9 @ A6 ) )
=> ( ( X
= ( F @ X9 ) )
=> ( B2
!= ( image_int_int @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1085_inj__img__insertE,axiom,
! [F: nat > int,A3: set_nat,X: int,B2: set_int] :
( ( inj_on_nat_int @ F @ A3 )
=> ( ~ ( member_int @ X @ B2 )
=> ( ( ( insert_int @ X @ B2 )
= ( image_nat_int @ F @ A3 ) )
=> ~ ! [X9: nat,A6: set_nat] :
( ~ ( member_nat @ X9 @ A6 )
=> ( ( A3
= ( insert_nat @ X9 @ A6 ) )
=> ( ( X
= ( F @ X9 ) )
=> ( B2
!= ( image_nat_int @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1086_inj__img__insertE,axiom,
! [F: int > nat,A3: set_int,X: nat,B2: set_nat] :
( ( inj_on_int_nat @ F @ A3 )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat @ X @ B2 )
= ( image_int_nat @ F @ A3 ) )
=> ~ ! [X9: int,A6: set_int] :
( ~ ( member_int @ X9 @ A6 )
=> ( ( A3
= ( insert_int @ X9 @ A6 ) )
=> ( ( X
= ( F @ X9 ) )
=> ( B2
!= ( image_int_nat @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1087_inj__img__insertE,axiom,
! [F: nat > nat,A3: set_nat,X: nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A3 )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat @ X @ B2 )
= ( image_nat_nat @ F @ A3 ) )
=> ~ ! [X9: nat,A6: set_nat] :
( ~ ( member_nat @ X9 @ A6 )
=> ( ( A3
= ( insert_nat @ X9 @ A6 ) )
=> ( ( X
= ( F @ X9 ) )
=> ( B2
!= ( image_nat_nat @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1088_inj__img__insertE,axiom,
! [F: set_int > int,A3: set_set_int,X: int,B2: set_int] :
( ( inj_on_set_int_int @ F @ A3 )
=> ( ~ ( member_int @ X @ B2 )
=> ( ( ( insert_int @ X @ B2 )
= ( image_set_int_int @ F @ A3 ) )
=> ~ ! [X9: set_int,A6: set_set_int] :
( ~ ( member_set_int @ X9 @ A6 )
=> ( ( A3
= ( insert_set_int @ X9 @ A6 ) )
=> ( ( X
= ( F @ X9 ) )
=> ( B2
!= ( image_set_int_int @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1089_inj__img__insertE,axiom,
! [F: int > set_int,A3: set_int,X: set_int,B2: set_set_int] :
( ( inj_on_int_set_int @ F @ A3 )
=> ( ~ ( member_set_int @ X @ B2 )
=> ( ( ( insert_set_int @ X @ B2 )
= ( image_int_set_int @ F @ A3 ) )
=> ~ ! [X9: int,A6: set_int] :
( ~ ( member_int @ X9 @ A6 )
=> ( ( A3
= ( insert_int @ X9 @ A6 ) )
=> ( ( X
= ( F @ X9 ) )
=> ( B2
!= ( image_int_set_int @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1090_inj__img__insertE,axiom,
! [F: nat > set_int,A3: set_nat,X: set_int,B2: set_set_int] :
( ( inj_on_nat_set_int @ F @ A3 )
=> ( ~ ( member_set_int @ X @ B2 )
=> ( ( ( insert_set_int @ X @ B2 )
= ( image_nat_set_int @ F @ A3 ) )
=> ~ ! [X9: nat,A6: set_nat] :
( ~ ( member_nat @ X9 @ A6 )
=> ( ( A3
= ( insert_nat @ X9 @ A6 ) )
=> ( ( X
= ( F @ X9 ) )
=> ( B2
!= ( image_nat_set_int @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1091_inj__img__insertE,axiom,
! [F: set_int > nat,A3: set_set_int,X: nat,B2: set_nat] :
( ( inj_on_set_int_nat @ F @ A3 )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat @ X @ B2 )
= ( image_set_int_nat @ F @ A3 ) )
=> ~ ! [X9: set_int,A6: set_set_int] :
( ~ ( member_set_int @ X9 @ A6 )
=> ( ( A3
= ( insert_set_int @ X9 @ A6 ) )
=> ( ( X
= ( F @ X9 ) )
=> ( B2
!= ( image_set_int_nat @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1092_inj__img__insertE,axiom,
! [F: set_set_int > int,A3: set_set_set_int,X: int,B2: set_int] :
( ( inj_on1358648829685248055nt_int @ F @ A3 )
=> ( ~ ( member_int @ X @ B2 )
=> ( ( ( insert_int @ X @ B2 )
= ( image_4671129442152397515nt_int @ F @ A3 ) )
=> ~ ! [X9: set_set_int,A6: set_set_set_int] :
( ~ ( member_set_set_int @ X9 @ A6 )
=> ( ( A3
= ( insert_set_set_int @ X9 @ A6 ) )
=> ( ( X
= ( F @ X9 ) )
=> ( B2
!= ( image_4671129442152397515nt_int @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1093_inj__img__insertE,axiom,
! [F: set_int > set_int,A3: set_set_int,X: set_int,B2: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ A3 )
=> ( ~ ( member_set_int @ X @ B2 )
=> ( ( ( insert_set_int @ X @ B2 )
= ( image_524474410958335435et_int @ F @ A3 ) )
=> ~ ! [X9: set_int,A6: set_set_int] :
( ~ ( member_set_int @ X9 @ A6 )
=> ( ( A3
= ( insert_set_int @ X9 @ A6 ) )
=> ( ( X
= ( F @ X9 ) )
=> ( B2
!= ( image_524474410958335435et_int @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_1094_inj__image__Compl__subset,axiom,
! [F: int > int,A3: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ord_less_eq_set_int @ ( image_int_int @ F @ ( uminus1532241313380277803et_int @ A3 ) ) @ ( uminus1532241313380277803et_int @ ( image_int_int @ F @ A3 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1095_inj__image__Compl__subset,axiom,
! [F: nat > int,A3: set_nat] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F @ ( uminus5710092332889474511et_nat @ A3 ) ) @ ( uminus1532241313380277803et_int @ ( image_nat_int @ F @ A3 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1096_inj__image__Compl__subset,axiom,
! [F: set_int > set_int,A3: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ top_top_set_set_int )
=> ( ord_le4403425263959731960et_int @ ( image_524474410958335435et_int @ F @ ( uminus7346710233107665121et_int @ A3 ) ) @ ( uminus7346710233107665121et_int @ ( image_524474410958335435et_int @ F @ A3 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1097_inj__image__Compl__subset,axiom,
! [F: int > set_int,A3: set_int] :
( ( inj_on_int_set_int @ F @ top_top_set_int )
=> ( ord_le4403425263959731960et_int @ ( image_int_set_int @ F @ ( uminus1532241313380277803et_int @ A3 ) ) @ ( uminus7346710233107665121et_int @ ( image_int_set_int @ F @ A3 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1098_inj__image__Compl__subset,axiom,
! [F: nat > set_int,A3: set_nat] :
( ( inj_on_nat_set_int @ F @ top_top_set_nat )
=> ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F @ ( uminus5710092332889474511et_nat @ A3 ) ) @ ( uminus7346710233107665121et_int @ ( image_nat_set_int @ F @ A3 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1099_inj__image__Compl__subset,axiom,
! [F: set_int > set_set_int,A3: set_set_int] :
( ( inj_on6285404204842837485et_int @ F @ top_top_set_set_int )
=> ( ord_le4317611570275147438et_int @ ( image_1010086626112315521et_int @ F @ ( uminus7346710233107665121et_int @ A3 ) ) @ ( uminus3047225438891465879et_int @ ( image_1010086626112315521et_int @ F @ A3 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1100_inj__image__Compl__subset,axiom,
! [F: int > set_set_int,A3: set_int] :
( ( inj_on3305102505822124087et_int @ F @ top_top_set_int )
=> ( ord_le4317611570275147438et_int @ ( image_6617583118289273547et_int @ F @ ( uminus1532241313380277803et_int @ A3 ) ) @ ( uminus3047225438891465879et_int @ ( image_6617583118289273547et_int @ F @ A3 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1101_inj__image__Compl__subset,axiom,
! [F: nat > set_set_int,A3: set_nat] :
( ( inj_on5614920437915075035et_int @ F @ top_top_set_nat )
=> ( ord_le4317611570275147438et_int @ ( image_8927401050382224495et_int @ F @ ( uminus5710092332889474511et_nat @ A3 ) ) @ ( uminus3047225438891465879et_int @ ( image_8927401050382224495et_int @ F @ A3 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1102_inj__image__subset__iff,axiom,
! [F: nat > int,A3: set_nat,B2: set_nat] :
( ( inj_on_nat_int @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_int @ ( image_nat_int @ F @ A3 ) @ ( image_nat_int @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_1103_inj__image__subset__iff,axiom,
! [F: int > int,A3: set_int,B2: set_int] :
( ( inj_on_int_int @ F @ top_top_set_int )
=> ( ( ord_less_eq_set_int @ ( image_int_int @ F @ A3 ) @ ( image_int_int @ F @ B2 ) )
= ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_1104_inj__image__subset__iff,axiom,
! [F: set_int > int,A3: set_set_int,B2: set_set_int] :
( ( inj_on_set_int_int @ F @ top_top_set_set_int )
=> ( ( ord_less_eq_set_int @ ( image_set_int_int @ F @ A3 ) @ ( image_set_int_int @ F @ B2 ) )
= ( ord_le4403425263959731960et_int @ A3 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_1105_inj__image__subset__iff,axiom,
! [F: nat > set_int,A3: set_nat,B2: set_nat] :
( ( inj_on_nat_set_int @ F @ top_top_set_nat )
=> ( ( ord_le4403425263959731960et_int @ ( image_nat_set_int @ F @ A3 ) @ ( image_nat_set_int @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_1106_inj__image__subset__iff,axiom,
! [F: int > set_int,A3: set_int,B2: set_int] :
( ( inj_on_int_set_int @ F @ top_top_set_int )
=> ( ( ord_le4403425263959731960et_int @ ( image_int_set_int @ F @ A3 ) @ ( image_int_set_int @ F @ B2 ) )
= ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_1107_inj__image__subset__iff,axiom,
! [F: set_set_int > int,A3: set_set_set_int,B2: set_set_set_int] :
( ( inj_on1358648829685248055nt_int @ F @ top_to5524576366173240574et_int )
=> ( ( ord_less_eq_set_int @ ( image_4671129442152397515nt_int @ F @ A3 ) @ ( image_4671129442152397515nt_int @ F @ B2 ) )
= ( ord_le4317611570275147438et_int @ A3 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_1108_inj__image__subset__iff,axiom,
! [F: set_int > set_int,A3: set_set_int,B2: set_set_int] :
( ( inj_on6435365835345961783et_int @ F @ top_top_set_set_int )
=> ( ( ord_le4403425263959731960et_int @ ( image_524474410958335435et_int @ F @ A3 ) @ ( image_524474410958335435et_int @ F @ B2 ) )
= ( ord_le4403425263959731960et_int @ A3 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_1109_inj__image__subset__iff,axiom,
! [F: nat > set_set_int,A3: set_nat,B2: set_nat] :
( ( inj_on5614920437915075035et_int @ F @ top_top_set_nat )
=> ( ( ord_le4317611570275147438et_int @ ( image_8927401050382224495et_int @ F @ A3 ) @ ( image_8927401050382224495et_int @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_1110_inj__image__subset__iff,axiom,
! [F: int > set_set_int,A3: set_int,B2: set_int] :
( ( inj_on3305102505822124087et_int @ F @ top_top_set_int )
=> ( ( ord_le4317611570275147438et_int @ ( image_6617583118289273547et_int @ F @ A3 ) @ ( image_6617583118289273547et_int @ F @ B2 ) )
= ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_1111_inj__image__subset__iff,axiom,
! [F: set_set_int > set_int,A3: set_set_set_int,B2: set_set_set_int] :
( ( inj_on8788328216580801005et_int @ F @ top_to5524576366173240574et_int )
=> ( ( ord_le4403425263959731960et_int @ ( image_3513010637850279041et_int @ F @ A3 ) @ ( image_3513010637850279041et_int @ F @ B2 ) )
= ( ord_le4317611570275147438et_int @ A3 @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_1112_inj__mult__left,axiom,
! [A: int] :
( ( inj_on_int_int @ ( times_times_int @ A ) @ top_top_set_int )
= ( A != zero_zero_int ) ) ).
% inj_mult_left
thf(fact_1113_zfact__iso__def,axiom,
( ring_zfact_iso
= ( ^ [P2: nat,K4: 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 @ P2 ) @ bot_bot_set_int ) ) @ ( semiri1314217659103216013at_int @ K4 ) ) ) ) ).
% zfact_iso_def
thf(fact_1114_the__elem__eq,axiom,
! [X: int] :
( ( the_elem_int @ ( insert_int @ X @ bot_bot_set_int ) )
= X ) ).
% the_elem_eq
thf(fact_1115_the__elem__eq,axiom,
! [X: set_int] :
( ( the_elem_set_int @ ( insert_set_int @ X @ bot_bot_set_set_int ) )
= X ) ).
% the_elem_eq
thf(fact_1116_UNIV__I,axiom,
! [X: set_int] : ( member_set_int @ X @ top_top_set_set_int ) ).
% UNIV_I
thf(fact_1117_UNIV__I,axiom,
! [X: int > set_int] : ( member_int_set_int @ X @ top_to3474959630705565203et_int ) ).
% UNIV_I
thf(fact_1118_UNIV__I,axiom,
! [X: set_set_int] : ( member_set_set_int @ X @ top_to5524576366173240574et_int ) ).
% UNIV_I
thf(fact_1119_UNIV__I,axiom,
! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% UNIV_I
thf(fact_1120_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_1121_subsetI,axiom,
! [A3: set_int_set_int,B2: set_int_set_int] :
( ! [X4: int > set_int] :
( ( member_int_set_int @ X4 @ A3 )
=> ( member_int_set_int @ X4 @ B2 ) )
=> ( ord_le2781113279154567107et_int @ A3 @ B2 ) ) ).
% subsetI
thf(fact_1122_subsetI,axiom,
! [A3: set_nat,B2: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_nat @ X4 @ B2 ) )
=> ( ord_less_eq_set_nat @ A3 @ B2 ) ) ).
% subsetI
thf(fact_1123_subsetI,axiom,
! [A3: set_int,B2: set_int] :
( ! [X4: int] :
( ( member_int @ X4 @ A3 )
=> ( member_int @ X4 @ B2 ) )
=> ( ord_less_eq_set_int @ A3 @ B2 ) ) ).
% subsetI
thf(fact_1124_subsetI,axiom,
! [A3: set_set_int,B2: set_set_int] :
( ! [X4: set_int] :
( ( member_set_int @ X4 @ A3 )
=> ( member_set_int @ X4 @ B2 ) )
=> ( ord_le4403425263959731960et_int @ A3 @ B2 ) ) ).
% subsetI
thf(fact_1125_subsetI,axiom,
! [A3: set_set_set_int,B2: set_set_set_int] :
( ! [X4: set_set_int] :
( ( member_set_set_int @ X4 @ A3 )
=> ( member_set_set_int @ X4 @ B2 ) )
=> ( ord_le4317611570275147438et_int @ A3 @ B2 ) ) ).
% subsetI
thf(fact_1126_subset__antisym,axiom,
! [A3: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A3 @ B2 )
=> ( ( ord_less_eq_set_int @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_1127_subset__antisym,axiom,
! [A3: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A3 @ B2 )
=> ( ( ord_le4403425263959731960et_int @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_1128_subset__antisym,axiom,
! [A3: set_set_set_int,B2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A3 @ B2 )
=> ( ( ord_le4317611570275147438et_int @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_1129_empty__iff,axiom,
! [C: int > set_int] :
~ ( member_int_set_int @ C @ bot_bo5203049024207062007et_int ) ).
% empty_iff
thf(fact_1130_empty__iff,axiom,
! [C: set_set_int] :
~ ( member_set_set_int @ C @ bot_bo2384636101374064866et_int ) ).
% empty_iff
thf(fact_1131_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_1132_empty__iff,axiom,
! [C: int] :
~ ( member_int @ C @ bot_bot_set_int ) ).
% empty_iff
thf(fact_1133_empty__iff,axiom,
! [C: set_int] :
~ ( member_set_int @ C @ bot_bot_set_set_int ) ).
% empty_iff
thf(fact_1134_all__not__in__conv,axiom,
! [A3: set_int_set_int] :
( ( ! [X5: int > set_int] :
~ ( member_int_set_int @ X5 @ A3 ) )
= ( A3 = bot_bo5203049024207062007et_int ) ) ).
% all_not_in_conv
thf(fact_1135_all__not__in__conv,axiom,
! [A3: set_set_set_int] :
( ( ! [X5: set_set_int] :
~ ( member_set_set_int @ X5 @ A3 ) )
= ( A3 = bot_bo2384636101374064866et_int ) ) ).
% all_not_in_conv
thf(fact_1136_all__not__in__conv,axiom,
! [A3: set_nat] :
( ( ! [X5: nat] :
~ ( member_nat @ X5 @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_1137_all__not__in__conv,axiom,
! [A3: set_int] :
( ( ! [X5: int] :
~ ( member_int @ X5 @ A3 ) )
= ( A3 = bot_bot_set_int ) ) ).
% all_not_in_conv
thf(fact_1138_all__not__in__conv,axiom,
! [A3: set_set_int] :
( ( ! [X5: set_int] :
~ ( member_set_int @ X5 @ A3 ) )
= ( A3 = bot_bot_set_set_int ) ) ).
% all_not_in_conv
thf(fact_1139_Collect__empty__eq,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( ! [X5: int] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_1140_Collect__empty__eq,axiom,
! [P: set_int > $o] :
( ( ( collect_set_int @ P )
= bot_bot_set_set_int )
= ( ! [X5: set_int] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_1141_empty__Collect__eq,axiom,
! [P: int > $o] :
( ( bot_bot_set_int
= ( collect_int @ P ) )
= ( ! [X5: int] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_1142_empty__Collect__eq,axiom,
! [P: set_int > $o] :
( ( bot_bot_set_set_int
= ( collect_set_int @ P ) )
= ( ! [X5: set_int] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_1143_image__eqI,axiom,
! [B: int,F: int > int,X: int,A3: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A3 )
=> ( member_int @ B @ ( image_int_int @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1144_image__eqI,axiom,
! [B: nat,F: int > nat,X: int,A3: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A3 )
=> ( member_nat @ B @ ( image_int_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1145_image__eqI,axiom,
! [B: int,F: nat > int,X: nat,A3: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A3 )
=> ( member_int @ B @ ( image_nat_int @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1146_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A3: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A3 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1147_image__eqI,axiom,
! [B: set_int,F: int > set_int,X: int,A3: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A3 )
=> ( member_set_int @ B @ ( image_int_set_int @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1148_image__eqI,axiom,
! [B: int,F: set_int > int,X: set_int,A3: set_set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_set_int @ X @ A3 )
=> ( member_int @ B @ ( image_set_int_int @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1149_image__eqI,axiom,
! [B: nat,F: set_int > nat,X: set_int,A3: set_set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_set_int @ X @ A3 )
=> ( member_nat @ B @ ( image_set_int_nat @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1150_image__eqI,axiom,
! [B: set_int,F: nat > set_int,X: nat,A3: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A3 )
=> ( member_set_int @ B @ ( image_nat_set_int @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1151_image__eqI,axiom,
! [B: set_set_int,F: int > set_set_int,X: int,A3: set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_int @ X @ A3 )
=> ( member_set_set_int @ B @ ( image_6617583118289273547et_int @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1152_image__eqI,axiom,
! [B: set_int,F: set_int > set_int,X: set_int,A3: set_set_int] :
( ( B
= ( F @ X ) )
=> ( ( member_set_int @ X @ A3 )
=> ( member_set_int @ B @ ( image_524474410958335435et_int @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_1153_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_1154_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_1155_insertCI,axiom,
! [A: int > set_int,B2: set_int_set_int,B: int > set_int] :
( ( ~ ( member_int_set_int @ A @ B2 )
=> ( A = B ) )
=> ( member_int_set_int @ A @ ( insert_int_set_int @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1156_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_1157_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1158_insert__iff,axiom,
! [A: int,B: int,A3: set_int] :
( ( member_int @ A @ ( insert_int @ B @ A3 ) )
= ( ( A = B )
| ( member_int @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_1159_insert__iff,axiom,
! [A: set_int,B: set_int,A3: set_set_int] :
( ( member_set_int @ A @ ( insert_set_int @ B @ A3 ) )
= ( ( A = B )
| ( member_set_int @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_1160_insert__iff,axiom,
! [A: int > set_int,B: int > set_int,A3: set_int_set_int] :
( ( member_int_set_int @ A @ ( insert_int_set_int @ B @ A3 ) )
= ( ( A = B )
| ( member_int_set_int @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_1161_insert__iff,axiom,
! [A: set_set_int,B: set_set_int,A3: set_set_set_int] :
( ( member_set_set_int @ A @ ( insert_set_set_int @ B @ A3 ) )
= ( ( A = B )
| ( member_set_set_int @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_1162_insert__iff,axiom,
! [A: nat,B: nat,A3: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A3 ) )
= ( ( A = B )
| ( member_nat @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_1163_insert__absorb2,axiom,
! [X: set_int,A3: set_set_int] :
( ( insert_set_int @ X @ ( insert_set_int @ X @ A3 ) )
= ( insert_set_int @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_1164_insert__absorb2,axiom,
! [X: int,A3: set_int] :
( ( insert_int @ X @ ( insert_int @ X @ A3 ) )
= ( insert_int @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_1165_DiffI,axiom,
! [C: int,A3: set_int,B2: set_int] :
( ( member_int @ C @ A3 )
=> ( ~ ( member_int @ C @ B2 )
=> ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1166_DiffI,axiom,
! [C: set_int,A3: set_set_int,B2: set_set_int] :
( ( member_set_int @ C @ A3 )
=> ( ~ ( member_set_int @ C @ B2 )
=> ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1167_DiffI,axiom,
! [C: int > set_int,A3: set_int_set_int,B2: set_int_set_int] :
( ( member_int_set_int @ C @ A3 )
=> ( ~ ( member_int_set_int @ C @ B2 )
=> ( member_int_set_int @ C @ ( minus_32553198013068476et_int @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1168_DiffI,axiom,
! [C: set_set_int,A3: set_set_set_int,B2: set_set_set_int] :
( ( member_set_set_int @ C @ A3 )
=> ( ~ ( member_set_set_int @ C @ B2 )
=> ( member_set_set_int @ C @ ( minus_6857623457997529383et_int @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1169_DiffI,axiom,
! [C: nat,A3: set_nat,B2: set_nat] :
( ( member_nat @ C @ A3 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_1170_Diff__iff,axiom,
! [C: int,A3: set_int,B2: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) )
= ( ( member_int @ C @ A3 )
& ~ ( member_int @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1171_Diff__iff,axiom,
! [C: set_int,A3: set_set_int,B2: set_set_int] :
( ( member_set_int @ C @ ( minus_8897228262479074673et_int @ A3 @ B2 ) )
= ( ( member_set_int @ C @ A3 )
& ~ ( member_set_int @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1172_Diff__iff,axiom,
! [C: int > set_int,A3: set_int_set_int,B2: set_int_set_int] :
( ( member_int_set_int @ C @ ( minus_32553198013068476et_int @ A3 @ B2 ) )
= ( ( member_int_set_int @ C @ A3 )
& ~ ( member_int_set_int @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1173_Diff__iff,axiom,
! [C: set_set_int,A3: set_set_set_int,B2: set_set_set_int] :
( ( member_set_set_int @ C @ ( minus_6857623457997529383et_int @ A3 @ B2 ) )
= ( ( member_set_set_int @ C @ A3 )
& ~ ( member_set_set_int @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1174_Diff__iff,axiom,
! [C: nat,A3: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
= ( ( member_nat @ C @ A3 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1175_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_1176_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_1177_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_1178_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_1179_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_1180_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_1181_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1182_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1183_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1184_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1185_mult__minus__right,axiom,
! [A: int,B: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_1186_minus__mult__minus,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
= ( times_times_int @ A @ B ) ) ).
% minus_mult_minus
thf(fact_1187_mult__minus__left,axiom,
! [A: int,B: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
= ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_1188_subset__empty,axiom,
! [A3: set_int] :
( ( ord_less_eq_set_int @ A3 @ bot_bot_set_int )
= ( A3 = bot_bot_set_int ) ) ).
% subset_empty
thf(fact_1189_subset__empty,axiom,
! [A3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A3 @ bot_bot_set_set_int )
= ( A3 = bot_bot_set_set_int ) ) ).
% subset_empty
thf(fact_1190_subset__empty,axiom,
! [A3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A3 @ bot_bo2384636101374064866et_int )
= ( A3 = bot_bo2384636101374064866et_int ) ) ).
% subset_empty
thf(fact_1191_empty__subsetI,axiom,
! [A3: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A3 ) ).
% empty_subsetI
thf(fact_1192_empty__subsetI,axiom,
! [A3: set_set_int] : ( ord_le4403425263959731960et_int @ bot_bot_set_set_int @ A3 ) ).
% empty_subsetI
thf(fact_1193_empty__subsetI,axiom,
! [A3: set_set_set_int] : ( ord_le4317611570275147438et_int @ bot_bo2384636101374064866et_int @ A3 ) ).
% empty_subsetI
thf(fact_1194_insert__subset,axiom,
! [X: int > set_int,A3: set_int_set_int,B2: set_int_set_int] :
( ( ord_le2781113279154567107et_int @ ( insert_int_set_int @ X @ A3 ) @ B2 )
= ( ( member_int_set_int @ X @ B2 )
& ( ord_le2781113279154567107et_int @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_1195_insert__subset,axiom,
! [X: nat,A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A3 ) @ B2 )
= ( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_1196_insert__subset,axiom,
! [X: int,A3: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ ( insert_int @ X @ A3 ) @ B2 )
= ( ( member_int @ X @ B2 )
& ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_1197_insert__subset,axiom,
! [X: set_int,A3: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( insert_set_int @ X @ A3 ) @ B2 )
= ( ( member_set_int @ X @ B2 )
& ( ord_le4403425263959731960et_int @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_1198_insert__subset,axiom,
! [X: set_set_int,A3: set_set_set_int,B2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ ( insert_set_set_int @ X @ A3 ) @ B2 )
= ( ( member_set_set_int @ X @ B2 )
& ( ord_le4317611570275147438et_int @ A3 @ B2 ) ) ) ).
% insert_subset
thf(fact_1199_image__empty,axiom,
! [F: int > int] :
( ( image_int_int @ F @ bot_bot_set_int )
= bot_bot_set_int ) ).
% image_empty
thf(fact_1200_image__empty,axiom,
! [F: int > set_int] :
( ( image_int_set_int @ F @ bot_bot_set_int )
= bot_bot_set_set_int ) ).
% image_empty
thf(fact_1201_image__empty,axiom,
! [F: set_int > set_set_int] :
( ( image_1010086626112315521et_int @ F @ bot_bot_set_set_int )
= bot_bo2384636101374064866et_int ) ).
% image_empty
thf(fact_1202_image__empty,axiom,
! [F: set_int > int] :
( ( image_set_int_int @ F @ bot_bot_set_set_int )
= bot_bot_set_int ) ).
% image_empty
thf(fact_1203_image__empty,axiom,
! [F: set_int > set_int] :
( ( image_524474410958335435et_int @ F @ bot_bot_set_set_int )
= bot_bot_set_set_int ) ).
% image_empty
thf(fact_1204_empty__is__image,axiom,
! [F: set_int > set_set_int,A3: set_set_int] :
( ( bot_bo2384636101374064866et_int
= ( image_1010086626112315521et_int @ F @ A3 ) )
= ( A3 = bot_bot_set_set_int ) ) ).
% empty_is_image
thf(fact_1205_empty__is__image,axiom,
! [F: int > int,A3: set_int] :
( ( bot_bot_set_int
= ( image_int_int @ F @ A3 ) )
= ( A3 = bot_bot_set_int ) ) ).
% empty_is_image
thf(fact_1206_empty__is__image,axiom,
! [F: set_int > int,A3: set_set_int] :
( ( bot_bot_set_int
= ( image_set_int_int @ F @ A3 ) )
= ( A3 = bot_bot_set_set_int ) ) ).
% empty_is_image
thf(fact_1207_empty__is__image,axiom,
! [F: int > set_int,A3: set_int] :
( ( bot_bot_set_set_int
= ( image_int_set_int @ F @ A3 ) )
= ( A3 = bot_bot_set_int ) ) ).
% empty_is_image
thf(fact_1208_empty__is__image,axiom,
! [F: set_int > set_int,A3: set_set_int] :
( ( bot_bot_set_set_int
= ( image_524474410958335435et_int @ F @ A3 ) )
= ( A3 = bot_bot_set_set_int ) ) ).
% empty_is_image
thf(fact_1209_image__is__empty,axiom,
! [F: set_int > set_set_int,A3: set_set_int] :
( ( ( image_1010086626112315521et_int @ F @ A3 )
= bot_bo2384636101374064866et_int )
= ( A3 = bot_bot_set_set_int ) ) ).
% image_is_empty
thf(fact_1210_image__is__empty,axiom,
! [F: int > int,A3: set_int] :
( ( ( image_int_int @ F @ A3 )
= bot_bot_set_int )
= ( A3 = bot_bot_set_int ) ) ).
% image_is_empty
thf(fact_1211_image__is__empty,axiom,
! [F: set_int > int,A3: set_set_int] :
( ( ( image_set_int_int @ F @ A3 )
= bot_bot_set_int )
= ( A3 = bot_bot_set_set_int ) ) ).
% image_is_empty
thf(fact_1212_image__is__empty,axiom,
! [F: int > set_int,A3: set_int] :
( ( ( image_int_set_int @ F @ A3 )
= bot_bot_set_set_int )
= ( A3 = bot_bot_set_int ) ) ).
% image_is_empty
thf(fact_1213_image__is__empty,axiom,
! [F: set_int > set_int,A3: set_set_int] :
( ( ( image_524474410958335435et_int @ F @ A3 )
= bot_bot_set_set_int )
= ( A3 = bot_bot_set_set_int ) ) ).
% image_is_empty
thf(fact_1214_singletonI,axiom,
! [A: int > set_int] : ( member_int_set_int @ A @ ( insert_int_set_int @ A @ bot_bo5203049024207062007et_int ) ) ).
% singletonI
thf(fact_1215_singletonI,axiom,
! [A: set_set_int] : ( member_set_set_int @ A @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) ).
% singletonI
thf(fact_1216_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_1217_singletonI,axiom,
! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% singletonI
thf(fact_1218_singletonI,axiom,
! [A: set_int] : ( member_set_int @ A @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ).
% singletonI
thf(fact_1219_insert__image,axiom,
! [X: int,A3: set_int,F: int > int] :
( ( member_int @ X @ A3 )
=> ( ( insert_int @ ( F @ X ) @ ( image_int_int @ F @ A3 ) )
= ( image_int_int @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1220_insert__image,axiom,
! [X: nat,A3: set_nat,F: nat > int] :
( ( member_nat @ X @ A3 )
=> ( ( insert_int @ ( F @ X ) @ ( image_nat_int @ F @ A3 ) )
= ( image_nat_int @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1221_insert__image,axiom,
! [X: int,A3: set_int,F: int > set_int] :
( ( member_int @ X @ A3 )
=> ( ( insert_set_int @ ( F @ X ) @ ( image_int_set_int @ F @ A3 ) )
= ( image_int_set_int @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1222_insert__image,axiom,
! [X: set_int,A3: set_set_int,F: set_int > int] :
( ( member_set_int @ X @ A3 )
=> ( ( insert_int @ ( F @ X ) @ ( image_set_int_int @ F @ A3 ) )
= ( image_set_int_int @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1223_insert__image,axiom,
! [X: nat,A3: set_nat,F: nat > set_int] :
( ( member_nat @ X @ A3 )
=> ( ( insert_set_int @ ( F @ X ) @ ( image_nat_set_int @ F @ A3 ) )
= ( image_nat_set_int @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1224_insert__image,axiom,
! [X: set_int,A3: set_set_int,F: set_int > set_int] :
( ( member_set_int @ X @ A3 )
=> ( ( insert_set_int @ ( F @ X ) @ ( image_524474410958335435et_int @ F @ A3 ) )
= ( image_524474410958335435et_int @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1225_insert__image,axiom,
! [X: set_set_int,A3: set_set_set_int,F: set_set_int > int] :
( ( member_set_set_int @ X @ A3 )
=> ( ( insert_int @ ( F @ X ) @ ( image_4671129442152397515nt_int @ F @ A3 ) )
= ( image_4671129442152397515nt_int @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1226_insert__image,axiom,
! [X: set_int,A3: set_set_int,F: set_int > set_set_int] :
( ( member_set_int @ X @ A3 )
=> ( ( insert_set_set_int @ ( F @ X ) @ ( image_1010086626112315521et_int @ F @ A3 ) )
= ( image_1010086626112315521et_int @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1227_insert__image,axiom,
! [X: int > set_int,A3: set_int_set_int,F: ( int > set_int ) > int] :
( ( member_int_set_int @ X @ A3 )
=> ( ( insert_int @ ( F @ X ) @ ( image_1876932965422846432nt_int @ F @ A3 ) )
= ( image_1876932965422846432nt_int @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1228_insert__image,axiom,
! [X: set_set_int,A3: set_set_set_int,F: set_set_int > set_int] :
( ( member_set_set_int @ X @ A3 )
=> ( ( insert_set_int @ ( F @ X ) @ ( image_3513010637850279041et_int @ F @ A3 ) )
= ( image_3513010637850279041et_int @ F @ A3 ) ) ) ).
% insert_image
thf(fact_1229_image__insert,axiom,
! [F: set_int > set_set_int,A: set_int,B2: set_set_int] :
( ( image_1010086626112315521et_int @ F @ ( insert_set_int @ A @ B2 ) )
= ( insert_set_set_int @ ( F @ A ) @ ( image_1010086626112315521et_int @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1230_image__insert,axiom,
! [F: set_int > set_int,A: set_int,B2: set_set_int] :
( ( image_524474410958335435et_int @ F @ ( insert_set_int @ A @ B2 ) )
= ( insert_set_int @ ( F @ A ) @ ( image_524474410958335435et_int @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1231_image__insert,axiom,
! [F: set_int > int,A: set_int,B2: set_set_int] :
( ( image_set_int_int @ F @ ( insert_set_int @ A @ B2 ) )
= ( insert_int @ ( F @ A ) @ ( image_set_int_int @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1232_image__insert,axiom,
! [F: int > set_int,A: int,B2: set_int] :
( ( image_int_set_int @ F @ ( insert_int @ A @ B2 ) )
= ( insert_set_int @ ( F @ A ) @ ( image_int_set_int @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1233_image__insert,axiom,
! [F: int > int,A: int,B2: set_int] :
( ( image_int_int @ F @ ( insert_int @ A @ B2 ) )
= ( insert_int @ ( F @ A ) @ ( image_int_int @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1234_Diff__empty,axiom,
! [A3: set_int] :
( ( minus_minus_set_int @ A3 @ bot_bot_set_int )
= A3 ) ).
% Diff_empty
thf(fact_1235_Diff__empty,axiom,
! [A3: set_set_int] :
( ( minus_8897228262479074673et_int @ A3 @ bot_bot_set_set_int )
= A3 ) ).
% Diff_empty
thf(fact_1236_empty__Diff,axiom,
! [A3: set_int] :
( ( minus_minus_set_int @ bot_bot_set_int @ A3 )
= bot_bot_set_int ) ).
% empty_Diff
thf(fact_1237_empty__Diff,axiom,
! [A3: set_set_int] :
( ( minus_8897228262479074673et_int @ bot_bot_set_set_int @ A3 )
= bot_bot_set_set_int ) ).
% empty_Diff
thf(fact_1238_Diff__cancel,axiom,
! [A3: set_int] :
( ( minus_minus_set_int @ A3 @ A3 )
= bot_bot_set_int ) ).
% Diff_cancel
thf(fact_1239_Diff__cancel,axiom,
! [A3: set_set_int] :
( ( minus_8897228262479074673et_int @ A3 @ A3 )
= bot_bot_set_set_int ) ).
% Diff_cancel
thf(fact_1240_insert__Diff1,axiom,
! [X: int,B2: set_int,A3: set_int] :
( ( member_int @ X @ B2 )
=> ( ( minus_minus_set_int @ ( insert_int @ X @ A3 ) @ B2 )
= ( minus_minus_set_int @ A3 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1241_insert__Diff1,axiom,
! [X: set_int,B2: set_set_int,A3: set_set_int] :
( ( member_set_int @ X @ B2 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X @ A3 ) @ B2 )
= ( minus_8897228262479074673et_int @ A3 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1242_insert__Diff1,axiom,
! [X: int > set_int,B2: set_int_set_int,A3: set_int_set_int] :
( ( member_int_set_int @ X @ B2 )
=> ( ( minus_32553198013068476et_int @ ( insert_int_set_int @ X @ A3 ) @ B2 )
= ( minus_32553198013068476et_int @ A3 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1243_insert__Diff1,axiom,
! [X: set_set_int,B2: set_set_set_int,A3: set_set_set_int] :
( ( member_set_set_int @ X @ B2 )
=> ( ( minus_6857623457997529383et_int @ ( insert_set_set_int @ X @ A3 ) @ B2 )
= ( minus_6857623457997529383et_int @ A3 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1244_insert__Diff1,axiom,
! [X: nat,B2: set_nat,A3: set_nat] :
( ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A3 ) @ B2 )
= ( minus_minus_set_nat @ A3 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1245_Diff__insert0,axiom,
! [X: int,A3: set_int,B2: set_int] :
( ~ ( member_int @ X @ A3 )
=> ( ( minus_minus_set_int @ A3 @ ( insert_int @ X @ B2 ) )
= ( minus_minus_set_int @ A3 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1246_Diff__insert0,axiom,
! [X: set_int,A3: set_set_int,B2: set_set_int] :
( ~ ( member_set_int @ X @ A3 )
=> ( ( minus_8897228262479074673et_int @ A3 @ ( insert_set_int @ X @ B2 ) )
= ( minus_8897228262479074673et_int @ A3 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1247_Diff__insert0,axiom,
! [X: int > set_int,A3: set_int_set_int,B2: set_int_set_int] :
( ~ ( member_int_set_int @ X @ A3 )
=> ( ( minus_32553198013068476et_int @ A3 @ ( insert_int_set_int @ X @ B2 ) )
= ( minus_32553198013068476et_int @ A3 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1248_Diff__insert0,axiom,
! [X: set_set_int,A3: set_set_set_int,B2: set_set_set_int] :
( ~ ( member_set_set_int @ X @ A3 )
=> ( ( minus_6857623457997529383et_int @ A3 @ ( insert_set_set_int @ X @ B2 ) )
= ( minus_6857623457997529383et_int @ A3 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1249_Diff__insert0,axiom,
! [X: nat,A3: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A3 )
=> ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ B2 ) )
= ( minus_minus_set_nat @ A3 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1250_psubsetI,axiom,
! [A3: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less_set_int @ A3 @ B2 ) ) ) ).
% psubsetI
thf(fact_1251_psubsetI,axiom,
! [A3: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less_set_set_int @ A3 @ B2 ) ) ) ).
% psubsetI
thf(fact_1252_psubsetI,axiom,
! [A3: set_set_set_int,B2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_le4562804192517611682et_int @ A3 @ B2 ) ) ) ).
% psubsetI
thf(fact_1253_Compl__anti__mono,axiom,
! [A3: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ A3 @ B2 )
=> ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B2 ) @ ( uminus1532241313380277803et_int @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_1254_Compl__anti__mono,axiom,
! [A3: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A3 @ B2 )
=> ( ord_le4403425263959731960et_int @ ( uminus7346710233107665121et_int @ B2 ) @ ( uminus7346710233107665121et_int @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_1255_Compl__anti__mono,axiom,
! [A3: set_set_set_int,B2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A3 @ B2 )
=> ( ord_le4317611570275147438et_int @ ( uminus3047225438891465879et_int @ B2 ) @ ( uminus3047225438891465879et_int @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_1256_Compl__subset__Compl__iff,axiom,
! [A3: set_int,B2: set_int] :
( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A3 ) @ ( uminus1532241313380277803et_int @ B2 ) )
= ( ord_less_eq_set_int @ B2 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1257_Compl__subset__Compl__iff,axiom,
! [A3: set_set_int,B2: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( uminus7346710233107665121et_int @ A3 ) @ ( uminus7346710233107665121et_int @ B2 ) )
= ( ord_le4403425263959731960et_int @ B2 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1258_Compl__subset__Compl__iff,axiom,
! [A3: set_set_set_int,B2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ ( uminus3047225438891465879et_int @ A3 ) @ ( uminus3047225438891465879et_int @ B2 ) )
= ( ord_le4317611570275147438et_int @ B2 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1259_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_1260_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_1261_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_1262_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_1263_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1264_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1265_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1266_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_1267_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_1268_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_1269_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_1270_r_Ocgenideal__minimal,axiom,
! [J: set_set_int,A: set_int] :
( ( ideal_7262958097527394045t_unit @ J @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
=> ( ( member_set_int @ A @ J )
=> ( ord_le4403425263959731960et_int @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A ) @ J ) ) ) ).
% r.cgenideal_minimal
thf(fact_1271_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_1272_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ x2 @ y2 )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ i @ ( semiri1314217659103216013at_int @ ( zfact_iso_inv @ n @ x2 ) ) ) @ ( 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 ) ) ) @ i @ ( semiri1314217659103216013at_int @ ( zfact_iso_inv @ n @ y2 ) ) ) ) ) ).
%------------------------------------------------------------------------------