%------------------------------------------------------------------------------
% 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_00147_005314__18288006_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1478 ( 547 unt; 204 typ;   0 def)
%            Number of atoms       : 3802 (1403 equ;   0 cnn)
%            Maximal formula atoms :    8 (   2 avg)
%            Number of connectives : 15634 ( 203   ~;  53   |; 184   &;13510   @)
%                                         (   0 <=>;1684  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   28 (   7 avg)
%            Number of types       :   21 (  20 usr)
%            Number of type conns  :  902 ( 902   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  186 ( 184 usr;  28 con; 0-4 aty)
%            Number of variables   : 3351 ( 209   ^;3028   !; 114   ?;3351   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:41:32.453
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_It__Int__Oint_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__Int__Oint_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J_J_J,type,
    partia4934656038542163276t_unit: $tType ).

thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Nat__Onat_Mt__Group__Omonoid__Omonoid____ext_It__Nat__Onat_Mt__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J_J_J,type,
    partia4692342223508353374t_unit: $tType ).

thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Int__Oint_Mt__Group__Omonoid__Omonoid____ext_It__Int__Oint_Mt__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J_J_J,type,
    partia2818514838349642498t_unit: $tType ).

thf(ty_n_t__Group__Omonoid__Omonoid____ext_It__Nat__Onat_Mt__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
    monoid5117177112048413924t_unit: $tType ).

thf(ty_n_t__Group__Omonoid__Omonoid____ext_It__Int__Oint_Mt__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J_J,type,
    monoid8431999971278595628t_unit: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Int__Oint_Mt__Set__Oset_It__Int__Oint_J_J_J_J,type,
    set_set_int_set_int: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
    set_set_set_set_int: $tType ).

thf(ty_n_t__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
    ring_e287156513554883748t_unit: $tType ).

thf(ty_n_t__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J,type,
    ring_e6626950497611839816t_unit: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Int__Oint_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
    set_int_set_int: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    set_set_set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
    set_set_set_int: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    set_set_int: $tType ).

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Ounit,type,
    product_unit: $tType ).

thf(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

thf(ty_n_t__Int__Oint,type,
    int: $tType ).

% Explicit typings (184)
thf(sy_c_AbelCoset_OA__RCOSETS_001t__Int__Oint_001t__Product____Type__Ounit,type,
    a_RCOS3445019769541752303t_unit: partia2818514838349642498t_unit > set_int > set_set_int ).

thf(sy_c_AbelCoset_OA__RCOSETS_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_AbelCoset_OA__SET__INV_001t__Int__Oint_001t__Product____Type__Ounit,type,
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thf(sy_c_AbelCoset_Oa__l__coset_001t__Int__Oint_001t__Product____Type__Ounit,type,
    a_l_co457318074463736598t_unit: partia2818514838349642498t_unit > int > set_int > set_int ).

thf(sy_c_AbelCoset_Oa__l__coset_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
    a_l_co3504123944629134560t_unit: partia4934656038542163276t_unit > set_int > set_set_int > set_set_int ).

thf(sy_c_AbelCoset_Oa__r__coset_001t__Int__Oint_001t__Product____Type__Ounit,type,
    a_r_co6205493800230438172t_unit: partia2818514838349642498t_unit > set_int > int > set_int ).

thf(sy_c_AbelCoset_Oa__r__coset_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
    a_r_co692709266861932262t_unit: partia4934656038542163276t_unit > set_set_int > set_int > set_set_int ).

thf(sy_c_AbelCoset_Oabelian__group__hom_001t__Int__Oint_001t__Product____Type__Ounit_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
    abelia4252751761703478237t_unit: partia2818514838349642498t_unit > partia4934656038542163276t_unit > ( int > set_int ) > $o ).

thf(sy_c_AbelCoset_Oabelian__subgroup_001t__Int__Oint_001t__Product____Type__Ounit,type,
    abelia4388627187832904468t_unit: set_int > partia2818514838349642498t_unit > $o ).

thf(sy_c_AbelCoset_Oadditive__subgroup_001t__Int__Oint_001t__Product____Type__Ounit,type,
    additi1413919225584036690t_unit: set_int > partia2818514838349642498t_unit > $o ).

thf(sy_c_AbelCoset_Oadditive__subgroup_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
    additi7073586575563672860t_unit: set_set_int > partia4934656038542163276t_unit > $o ).

thf(sy_c_AbelCoset_Oset__add_001t__Int__Oint_001t__Product____Type__Ounit,type,
    set_ad6660713454305125854t_unit: partia2818514838349642498t_unit > set_int > set_int > set_int ).

thf(sy_c_AbelCoset_Oset__add_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
    set_ad273131178244904872t_unit: partia4934656038542163276t_unit > set_set_int > set_set_int > set_set_int ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Int__Oint_Mt__Set__Oset_It__Int__Oint_J_J,type,
    comple465308567373470592et_int: set_int_set_int > int > set_int ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
    complete_Sup_Sup_o: set_o > $o ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
    complete_Sup_Sup_int: set_int > int ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
    complete_Sup_Sup_nat: set_nat > nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_It__Int__Oint_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
    comple8192055500899288118et_int: set_set_int_set_int > set_int_set_int ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Int__Oint_J,type,
    comple3221217463730067765et_int: set_set_int > set_int ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7399068483239264473et_nat: set_set_nat > set_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    comple7281953568134767595et_int: set_set_set_int > set_set_int ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
    comple1756060948637632417et_int: set_set_set_set_int > set_set_set_int ).

thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Int__Oint_001t__Group__Omonoid__Omonoid____ext_It__Int__Oint_Mt__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J_J,type,
    partia8426541738980984321t_unit: partia2818514838349642498t_unit > set_int ).

thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Nat__Onat_001t__Group__Omonoid__Omonoid____ext_It__Nat__Onat_Mt__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
    partia3499330772048238685t_unit: partia4692342223508353374t_unit > set_nat ).

thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Set__Oset_It__Int__Oint_J_001t__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__Int__Oint_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J_J,type,
    partia966996272515721803t_unit: partia4934656038542163276t_unit > set_set_int ).

thf(sy_c_Congruence_Opartial__object_Opartial__object__ext_001t__Int__Oint_001t__Group__Omonoid__Omonoid____ext_It__Int__Oint_Mt__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J_J,type,
    partia4118392927963588428t_unit: set_int > monoid8431999971278595628t_unit > partia2818514838349642498t_unit ).

thf(sy_c_Congruence_Opartial__object_Opartial__object__ext_001t__Nat__Onat_001t__Group__Omonoid__Omonoid____ext_It__Nat__Onat_Mt__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
    partia8414553997885618600t_unit: set_nat > monoid5117177112048413924t_unit > partia4692342223508353374t_unit ).

thf(sy_c_Coset_Oorder_001t__Set__Oset_It__Int__Oint_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J,type,
    order_4716970363388151434t_unit: partia4934656038542163276t_unit > nat ).

thf(sy_c_Field_Omod__ring,type,
    mod_ring: nat > partia4692342223508353374t_unit ).

thf(sy_c_Field_Ozfact__iso__inv,type,
    zfact_iso_inv: nat > set_int > nat ).

thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
    finite_finite_int: set_int > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
    finite6197958912794628473et_int: set_set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Int__Oint,type,
    inj_on_int_int: ( int > int ) > set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Nat__Onat,type,
    inj_on_int_nat: ( int > nat ) > set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
    inj_on_int_set_int: ( int > set_int ) > set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    inj_on3305102505822124087et_int: ( int > set_set_int ) > set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Int__Oint,type,
    inj_on_nat_int: ( nat > int ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
    inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
    inj_on_nat_set_int: ( nat > set_int ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    inj_on_nat_set_nat: ( nat > set_nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    inj_on5614920437915075035et_int: ( nat > set_set_int ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Int__Oint_J_001t__Int__Oint,type,
    inj_on_set_int_int: ( set_int > int ) > set_set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Int__Oint_J_001t__Nat__Onat,type,
    inj_on_set_int_nat: ( set_int > nat ) > set_set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
    inj_on6435365835345961783et_int: ( set_int > set_int ) > set_set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    inj_on6285404204842837485et_int: ( set_int > set_set_int ) > set_set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
    inj_on5116562847525634467et_int: ( set_int > set_set_set_int ) > set_set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    inj_on2776966659131765557et_nat: ( set_nat > set_set_nat ) > set_set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Nat__Onat,type,
    inj_on1361139300194298331nt_nat: ( set_set_int > nat ) > set_set_set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    inj_on6097848081577782435et_int: ( set_set_int > set_set_int ) > set_set_set_int > $o ).

thf(sy_c_Group_Omonoid_Omonoid__ext_001t__Int__Oint_001t__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J,type,
    monoid6080699973261426200t_unit: ( int > int > int ) > int > ring_e6626950497611839816t_unit > monoid8431999971278595628t_unit ).

thf(sy_c_Group_Omonoid_Omonoid__ext_001t__Nat__Onat_001t__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
    monoid7204241362306134736t_unit: ( nat > nat > nat ) > nat > ring_e287156513554883748t_unit > monoid5117177112048413924t_unit ).

thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
    one_one_int: int ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
    plus_plus_int: int > int > int ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
    times_times_int: int > int > int ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
    times_times_nat: nat > nat > nat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
    uminus_uminus_int: int > int ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
    uminus1532241313380277803et_int: set_int > set_int ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
    uminus5710092332889474511et_nat: set_nat > set_nat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    uminus7346710233107665121et_int: set_set_int > set_set_int ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
    zero_zero_int: int ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_Ideal_Ogenideal_001t__Int__Oint_001t__Product____Type__Ounit,type,
    genide1613390280493775889t_unit: partia2818514838349642498t_unit > set_int > set_int ).

thf(sy_c_Ideal_Oprincipalideal_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
    princi8860937869964495385t_unit: set_set_int > partia4934656038542163276t_unit > $o ).

thf(sy_c_IntRing_OZFact,type,
    zFact: int > partia4934656038542163276t_unit ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: nat > int ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: nat > nat ).

thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
    bot_bot_o: $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Int__Oint_Mt__Set__Oset_It__Int__Oint_J_J_J,type,
    bot_bo5203049024207062007et_int: set_int_set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
    bot_bot_set_o: set_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    bot_bot_set_set_int: set_set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
    bot_bo2384636101374064866et_int: set_set_set_int ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Int__Oint_Mt__Set__Oset_It__Int__Oint_J_J,type,
    ord_less_int_set_int: ( int > set_int ) > ( int > set_int ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
    ord_less_o: $o > $o > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
    ord_less_int: int > int > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_set_int: set_int > set_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    ord_less_set_set_int: set_set_int > set_set_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
    ord_le4562804192517611682et_int: set_set_set_int > set_set_set_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_Mt__Set__Oset_It__Int__Oint_J_J,type,
    ord_le4076609313706656525et_int: ( int > set_int ) > ( int > set_int ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
    ord_less_eq_o: $o > $o > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: int > int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
    ord_less_eq_set_o: set_o > set_o > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_eq_set_int: set_int > set_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
    ord_le4317611570275147438et_int: set_set_set_int > set_set_set_int > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
    top_top_o: $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    top_top_set_set_int: set_set_int ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    top_top_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
    top_to5524576366173240574et_int: set_set_set_int ).

thf(sy_c_Product__Type_OUnity,type,
    product_Unity: product_unit ).

thf(sy_c_QuotRing_Orcoset__mult_001t__Int__Oint_001t__Product____Type__Ounit,type,
    rcoset2619718275356395046t_unit: partia2818514838349642498t_unit > set_int > set_int > set_int > set_int ).

thf(sy_c_Ring_Oa__inv_001t__Int__Oint_001t__Product____Type__Ounit,type,
    a_inv_8811962894454695315t_unit: partia2818514838349642498t_unit > int > int ).

thf(sy_c_Ring_Oa__inv_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
    a_inv_5951419416477254493t_unit: partia4934656038542163276t_unit > set_int > set_int ).

thf(sy_c_Ring_Ofield_001t__Int__Oint_001t__Product____Type__Ounit,type,
    field_5117527561578272769t_unit: partia2818514838349642498t_unit > $o ).

thf(sy_c_Ring_Oring_Oadd_001t__Int__Oint_001t__Product____Type__Ounit,type,
    add_int_Product_unit: partia2818514838349642498t_unit > int > int > int ).

thf(sy_c_Ring_Oring_Oadd_001t__Nat__Onat_001t__Product____Type__Ounit,type,
    add_nat_Product_unit: partia4692342223508353374t_unit > nat > nat > nat ).

thf(sy_c_Ring_Oring_Oadd_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
    add_se5859248395121729892t_unit: partia4934656038542163276t_unit > set_int > set_int > set_int ).

thf(sy_c_Ring_Oring_Oring__ext_001t__Int__Oint_001t__Product____Type__Ounit,type,
    ring_e5272872978682396362t_unit: int > ( int > int > int ) > product_unit > ring_e6626950497611839816t_unit ).

thf(sy_c_Ring_Oring_Oring__ext_001t__Nat__Onat_001t__Product____Type__Ounit,type,
    ring_e8156451031480216102t_unit: nat > ( nat > nat > nat ) > product_unit > ring_e287156513554883748t_unit ).

thf(sy_c_Ring_Oring_Ozero_001t__Int__Oint_001t__Product____Type__Ounit,type,
    zero_i2266321264637750939t_unit: partia2818514838349642498t_unit > int ).

thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
    zero_s6269048424454532197t_unit: partia4934656038542163276t_unit > set_int ).

thf(sy_c_Ring_Oring__hom_001t__Int__Oint_001t__Product____Type__Ounit_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
    ring_h1577560673974712708t_unit: partia2818514838349642498t_unit > partia4934656038542163276t_unit > set_int_set_int ).

thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: int > int > int ).

thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
    modulo_modulo_nat: nat > nat > nat ).

thf(sy_c_Set_OCollect_001_062_It__Int__Oint_Mt__Set__Oset_It__Int__Oint_J_J,type,
    collect_int_set_int: ( ( int > set_int ) > $o ) > set_int_set_int ).

thf(sy_c_Set_OCollect_001t__Int__Oint,type,
    collect_int: ( int > $o ) > set_int ).

thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
    collect_set_int: ( set_int > $o ) > set_set_int ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    collect_set_set_int: ( set_set_int > $o ) > set_set_set_int ).

thf(sy_c_Set_Oimage_001_062_It__Int__Oint_Mt__Set__Oset_It__Int__Oint_J_J_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    image_3533601879074506316et_int: ( ( int > set_int ) > set_set_int ) > set_int_set_int > set_set_set_int ).

thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
    image_o_o: ( $o > $o ) > set_o > set_o ).

thf(sy_c_Set_Oimage_001t__Int__Oint_001_Eo,type,
    image_int_o: ( int > $o ) > set_int > set_o ).

thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
    image_int_int: ( int > int ) > set_int > set_int ).

thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
    image_int_nat: ( int > nat ) > set_int > set_nat ).

thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
    image_int_set_int: ( int > set_int ) > set_int > set_set_int ).

thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J,type,
    image_int_set_nat: ( int > set_nat ) > set_int > set_set_nat ).

thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    image_6617583118289273547et_int: ( int > set_set_int ) > set_int > set_set_set_int ).

thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
    image_63215931098211969et_int: ( int > set_set_set_int ) > set_int > set_set_set_set_int ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
    image_nat_o: ( nat > $o ) > set_nat > set_o ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
    image_nat_int: ( nat > int ) > set_nat > set_int ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Int__Oint_J,type,
    image_nat_set_int: ( nat > set_int ) > set_nat > set_set_int ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    image_8927401050382224495et_int: ( nat > set_set_int ) > set_nat > set_set_set_int ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    image_2194112158459175443et_nat: ( nat > set_set_nat ) > set_nat > set_set_set_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
    image_924495994448699429et_int: ( nat > set_set_set_int ) > set_nat > set_set_set_set_int ).

thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Int__Oint_Mt__Set__Oset_It__Int__Oint_J_J_J_001_Eo,type,
    image_5276478557982445742_int_o: ( set_int_set_int > $o ) > set_set_int_set_int > set_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001_Eo,type,
    image_set_int_o: ( set_int > $o ) > set_set_int > set_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Int__Oint,type,
    image_set_int_int: ( set_int > int ) > set_set_int > set_int ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Nat__Onat,type,
    image_set_int_nat: ( set_int > nat ) > set_set_int > set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
    image_524474410958335435et_int: ( set_int > set_int ) > set_set_int > set_set_int ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Nat__Onat_J,type,
    image_4702325430467532143et_nat: ( set_int > set_nat ) > set_set_int > set_set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    image_1010086626112315521et_int: ( set_int > set_set_int ) > set_set_int > set_set_set_int ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
    image_1737623724905622583et_int: ( set_int > set_set_set_int ) > set_set_int > set_set_set_set_int ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
    image_set_nat_o: ( set_nat > $o ) > set_set_nat > set_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Int__Oint_J,type,
    image_3739036796817536367et_int: ( set_nat > set_int ) > set_set_nat > set_set_int ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    image_4234937972324292645et_int: ( set_nat > set_set_int ) > set_set_nat > set_set_set_int ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001_Eo,type,
    image_set_set_int_o: ( set_set_int > $o ) > set_set_set_int > set_o ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Nat__Onat,type,
    image_4673619912661447791nt_nat: ( set_set_int > nat ) > set_set_set_int > set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Set__Oset_It__Int__Oint_J,type,
    image_3513010637850279041et_int: ( set_set_int > set_int ) > set_set_set_int > set_set_int ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
    image_7690861657359475749et_nat: ( set_set_int > set_nat ) > set_set_set_int > set_set_nat ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    image_2718908958957770551et_int: ( set_set_int > set_set_int ) > set_set_set_int > set_set_set_int ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J_001_Eo,type,
    image_2622566851371609091_int_o: ( set_set_set_int > $o ) > set_set_set_set_int > set_o ).

thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
    insert_int: int > set_int > set_int ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Int__Oint_J,type,
    insert_set_int: set_int > set_set_int > set_set_int ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001_062_It__Int__Oint_Mt__Set__Oset_It__Int__Oint_J_J,type,
    set_or7631710232683477194et_int: ( int > set_int ) > set_int_set_int ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
    set_ord_lessThan_int: int > set_int ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Int__Oint_J,type,
    set_or5935648273017318783et_int: set_int > set_set_int ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    set_or4141871561713569845et_int: set_set_int > set_set_set_int ).

thf(sy_c_UnivPoly_Obound_001t__Set__Oset_It__Int__Oint_J,type,
    bound_set_int: set_int > nat > ( nat > set_int ) > $o ).

thf(sy_c_member_001_062_It__Int__Oint_Mt__Set__Oset_It__Int__Oint_J_J,type,
    member_int_set_int: ( int > set_int ) > set_int_set_int > $o ).

thf(sy_c_member_001_Eo,type,
    member_o: $o > set_o > $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_n__ge__1,axiom,
    ord_less_nat @ one_one_nat @ n ).

% n_ge_1
thf(fact_6_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_7_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_8_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_9_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_10_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_11_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_12__092_060open_062I_A_L_062_092_060_094bsub_062_092_060Z_062_092_060_094esub_062_Aint_Ax_H_A_L_Aint_Ay_H_A_061_AI_A_L_062_092_060_094bsub_062_092_060Z_062_092_060_094esub_062_Aint_A_I_Ix_H_A_L_Ay_H_J_Amod_An_J_092_060close_062,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 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ x ) @ ( semiri1314217659103216013at_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 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( plus_plus_nat @ x @ y ) @ n ) ) ) ) ).

% \<open>I +>\<^bsub>\<Z>\<^esub> int x' + int y' = I +>\<^bsub>\<Z>\<^esub> int ((x' + y') mod n)\<close>
thf(fact_13_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_14_mod__ring__def,axiom,
    ( mod_ring
    = ( ^ [N: nat] :
          ( partia8414553997885618600t_unit @ ( set_ord_lessThan_nat @ N )
          @ ( monoid7204241362306134736t_unit
            @ ^ [X2: nat,Y2: nat] : ( modulo_modulo_nat @ ( times_times_nat @ X2 @ Y2 ) @ N )
            @ one_one_nat
            @ ( ring_e8156451031480216102t_unit @ zero_zero_nat
              @ ^ [X2: nat,Y2: nat] : ( modulo_modulo_nat @ ( plus_plus_nat @ X2 @ Y2 ) @ N )
              @ product_Unity ) ) ) ) ) ).

% mod_ring_def
thf(fact_15_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_16_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_17_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_18_i_Oa__coset__eq,axiom,
    ! [X3: int] :
      ( ( member_int @ X3 @ 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 @ X3 )
        = ( 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 ) ) ) @ X3 @ i ) ) ) ).

% i.a_coset_eq
thf(fact_19_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_20_i_Oa__rcos__module,axiom,
    ! [X: int,X4: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( member_int @ X4 @ top_top_set_int )
       => ( ( member_int @ X4 @ ( 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 @ X4 @ ( uminus_uminus_int @ X ) ) @ i ) ) ) ) ).

% i.a_rcos_module
thf(fact_21_i_Oa__rcos__module__imp,axiom,
    ! [X: int,X4: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( member_int @ X4 @ ( 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 @ X4 @ ( uminus_uminus_int @ X ) ) @ i ) ) ) ).

% i.a_rcos_module_imp
thf(fact_22_i_Oa__rcos__module__rev,axiom,
    ! [X: int,X4: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( member_int @ X4 @ top_top_set_int )
       => ( ( member_int @ ( plus_plus_int @ X4 @ ( uminus_uminus_int @ X ) ) @ i )
         => ( member_int @ X4 @ ( 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_23_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_24_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_mult
thf(fact_25_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_mult
thf(fact_26_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self1
thf(fact_27_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self1
thf(fact_28_n__ge__0,axiom,
    ord_less_nat @ zero_zero_nat @ n ).

% n_ge_0
thf(fact_29_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_30_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_31_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_32_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_33_mod__mod__trivial,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_34_mod__add__self2,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self2
thf(fact_35_mod__add__self2,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self2
thf(fact_36_mod__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self1
thf(fact_37_mod__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self1
thf(fact_38_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_39_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_40_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_41_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_42_add__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N2 = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_43_mod__minus__minus,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_minus_minus
thf(fact_44_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% nat_add_left_cancel_less
thf(fact_45_mult__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N2 @ K ) )
      = ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_46_mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_47_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_48_mult__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_49_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_50_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_51_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_52_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_53_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_55_Collect__mem__eq,axiom,
    ! [A3: set_set_set_int] :
      ( ( collect_set_set_int
        @ ^ [X2: set_set_int] : ( member_set_set_int @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
    ! [A3: set_set_int] :
      ( ( collect_set_int
        @ ^ [X2: set_int] : ( member_set_int @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
    ! [A3: set_int_set_int] :
      ( ( collect_int_set_int
        @ ^ [X2: int > set_int] : ( member_int_set_int @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
    ! [A3: set_int] :
      ( ( collect_int
        @ ^ [X2: int] : ( member_int @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_60_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X5: int] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_61_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X5: nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_62_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N2 ) )
      = ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_63_mod__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = M ) ) ).

% mod_less
thf(fact_64_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_65_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_66_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_67_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_68_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_69_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_70_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_71_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_72_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_add
thf(fact_73_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_add
thf(fact_74_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N2 )
        = one_one_nat )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_75_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ N2 )
        = one_one_int )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_76_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_77_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_78_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_79_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_80_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_81_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_82_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_83_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_84_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self4
thf(fact_85_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self4
thf(fact_86_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self3
thf(fact_87_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self3
thf(fact_88_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self2
thf(fact_89_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self2
thf(fact_90_add__gr__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% add_gr_0
thf(fact_91_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_92_nat__0__less__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% nat_0_less_mult_iff
thf(fact_93_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% mult_less_cancel2
thf(fact_94_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_95_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_96_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_97_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_98_i_Oa__setinv__closed,axiom,
    ! [K2: set_int] :
      ( ( member_set_int @ K2 @ ( 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 ) ) ) @ K2 ) @ ( 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_99_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_100_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_101_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_102_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_103_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_104_nat__neq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( M != N2 )
      = ( ( ord_less_nat @ M @ N2 )
        | ( ord_less_nat @ N2 @ M ) ) ) ).

% nat_neq_iff
thf(fact_105_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_106_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( M != N2 ) ) ).

% less_not_refl2
thf(fact_107_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_108_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_109_nat__less__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
             => ( P @ M2 ) )
         => ( P @ N3 ) )
     => ( P @ N2 ) ) ).

% nat_less_induct
thf(fact_110_infinite__descent,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ~ ( P @ N3 )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
              & ~ ( P @ M2 ) ) )
     => ( P @ N2 ) ) ).

% infinite_descent
thf(fact_111_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_112_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L )
       != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
thf(fact_113_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L )
       != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
thf(fact_114_infinite__descent0,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P @ N3 )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N3 )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N2 ) ) ) ).

% infinite_descent0
thf(fact_115_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_116_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_117_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_118_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_119_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_120_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_121_mod__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% mod_minus_right
thf(fact_122_mod__minus__cong,axiom,
    ! [A: int,B: int,A2: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = ( modulo_modulo_int @ A2 @ B ) )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
        = ( modulo_modulo_int @ ( uminus_uminus_int @ A2 ) @ B ) ) ) ).

% mod_minus_cong
thf(fact_123_mod__minus__eq,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% mod_minus_eq
thf(fact_124_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% less_add_eq_less
thf(fact_125_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_126_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_127_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_128_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_129_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_130_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_131_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_132_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_133_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_134_zmod__int,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% zmod_int
thf(fact_135_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K3 )
          & ( ( plus_plus_nat @ I @ K3 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_136_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_137_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_138_mod__less__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).

% mod_less_divisor
thf(fact_139_split__mod,axiom,
    ! [Q: nat > $o,M: nat,N2: nat] :
      ( ( Q @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( Q @ M ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I2: nat,J2: nat] :
              ( ( ( ord_less_nat @ J2 @ N2 )
                & ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I2 ) @ J2 ) ) )
             => ( Q @ J2 ) ) ) ) ) ).

% split_mod
thf(fact_140_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_141_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_142_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_143_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_144_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_145_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_146_mod__add__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A2 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B2 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_147_mod__add__cong,axiom,
    ! [A: int,C: int,A2: int,B: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A2 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B2 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_148_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_149_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_150_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_151_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_152_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_153_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_154_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_155_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] :
      ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
      = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_156_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
      = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_157_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_158_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A2 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B2 @ C ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B2 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_159_mod__mult__cong,axiom,
    ! [A: int,C: int,A2: int,B: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A2 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B2 @ C ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( times_times_int @ A2 @ B2 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_160_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_161_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_162_add__eq__self__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = M )
     => ( N2 = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_163_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_164_mult__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% mult_0
thf(fact_165_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% add_mult_distrib2
thf(fact_166_add__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% add_mult_distrib
thf(fact_167_nat__mult__1__right,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ N2 @ one_one_nat )
      = N2 ) ).

% nat_mult_1_right
thf(fact_168_nat__mult__1,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ one_one_nat @ N2 )
      = N2 ) ).

% nat_mult_1
thf(fact_169_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
     => ~ ! [D: int] :
            ( B
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D ) ) ) ) ).

% mod_eqE
thf(fact_170_mult__eq__self__implies__10,axiom,
    ! [M: nat,N2: nat] :
      ( ( M
        = ( times_times_nat @ M @ N2 ) )
     => ( ( N2 = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_171_nat__mod__eq__iff,axiom,
    ! [X: nat,N2: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N2 )
        = ( modulo_modulo_nat @ Y @ N2 ) )
      = ( ? [Q1: nat,Q2: nat] :
            ( ( plus_plus_nat @ X @ ( times_times_nat @ N2 @ Q1 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_172_int__Zcarr,axiom,
    ! [K: int] : ( member_int @ K @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ).

% int_Zcarr
thf(fact_173_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_174_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_175_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_176_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_177_i_Oa__rcosets__carrier,axiom,
    ! [X6: set_int] :
      ( ( member_set_int @ X6 @ ( 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 @ X6 @ top_top_set_int ) ) ).

% i.a_rcosets_carrier
thf(fact_178_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_179_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_180_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 ) ) ) ) )
     => ? [X5: int] :
          ( ( member_int @ X5 @ ( 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 @ X5 )
            = ( 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_181_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 ) ) ) ) )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ ( 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 @ X5 )
              = X5 ) )
       => ( 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_182_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 ) ) ) ) )
     => ? [X5: int] :
          ( ( member_int @ X5 @ ( 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 ) ) ) @ X5 @ 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_183_UNIV_I4_J,axiom,
    ! [P: int > $o] :
      ( ( ? [X2: int] :
            ( ( member_int @ X2 @ top_top_set_int )
            & ( P @ X2 ) ) )
      = ( ? [X7: int] : ( P @ X7 ) ) ) ).

% UNIV(4)
thf(fact_184_UNIV_I4_J,axiom,
    ! [P: nat > $o] :
      ( ( ? [X2: nat] :
            ( ( member_nat @ X2 @ top_top_set_nat )
            & ( P @ X2 ) ) )
      = ( ? [X7: nat] : ( P @ X7 ) ) ) ).

% UNIV(4)
thf(fact_185_UNIV_I3_J,axiom,
    ! [P: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int @ X2 @ top_top_set_int )
           => ( P @ X2 ) ) )
      = ( ! [X7: int] : ( P @ X7 ) ) ) ).

% UNIV(3)
thf(fact_186_UNIV_I3_J,axiom,
    ! [P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ top_top_set_nat )
           => ( P @ X2 ) ) )
      = ( ! [X7: nat] : ( P @ X7 ) ) ) ).

% UNIV(3)
thf(fact_187_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_188_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_189_int_Oadd_Oone__closed,axiom,
    member_int @ zero_zero_int @ top_top_set_int ).

% int.add.one_closed
thf(fact_190_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_191_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_192_int_Oone__closed,axiom,
    member_int @ one_one_int @ top_top_set_int ).

% int.one_closed
thf(fact_193_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_194_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_195_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_196_int_Ominus__zero,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% int.minus_zero
thf(fact_197_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_198_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_199_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_200_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_201_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_202_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_203_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_204_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_205_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_206_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_207_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_208_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_209_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_210_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_211_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_212_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_213_i_Oadd_Oinv__unique,axiom,
    ! [Y: int,X: int,Y3: 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 @ Y3 )
          = ( 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 @ Y3 @ ( 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 = Y3 ) ) ) ) ) ) ).

% i.add.inv_unique
thf(fact_214_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_215_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_216_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_217_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_218_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_219_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_220_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_221_int_Oa__coset__add__zero,axiom,
    ! [M3: set_int] :
      ( ( ord_less_eq_set_int @ M3 @ 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 ) ) ) @ M3 @ zero_zero_int )
        = M3 ) ) ).

% int.a_coset_add_zero
thf(fact_222_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_223_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_224_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_225_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_226_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_227_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_228_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_229_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).

% of_nat_mono
thf(fact_230_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).

% of_nat_mono
thf(fact_231_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_232_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_233_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_234_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_235_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_236_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_237_UNIV_I2_J,axiom,
    ! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ top_top_set_nat ) ).

% UNIV(2)
thf(fact_238_UNIV_I2_J,axiom,
    ! [A3: set_int] : ( ord_less_eq_set_int @ A3 @ top_top_set_int ) ).

% UNIV(2)
thf(fact_239_UNIV_I2_J,axiom,
    ! [A3: set_set_int] : ( ord_le4403425263959731960et_int @ A3 @ top_top_set_set_int ) ).

% UNIV(2)
thf(fact_240_int_Oadd_Ocarrier__not__empty,axiom,
    top_top_set_int != bot_bot_set_int ).

% int.add.carrier_not_empty
thf(fact_241_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).

% Euclidean_Division.pos_mod_bound
thf(fact_242_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).

% neg_mod_bound
thf(fact_243_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_244_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_245_int_Oadd_Oone__in__subset,axiom,
    ! [H2: set_int] :
      ( ( ord_less_eq_set_int @ H2 @ top_top_set_int )
     => ( ( H2 != bot_bot_set_int )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ H2 )
             => ( member_int @ ( uminus_uminus_int @ X5 ) @ H2 ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ H2 )
               => ! [Xa: int] :
                    ( ( member_int @ Xa @ H2 )
                   => ( member_int @ ( plus_plus_int @ X5 @ Xa ) @ H2 ) ) )
           => ( member_int @ zero_zero_int @ H2 ) ) ) ) ) ).

% int.add.one_in_subset
thf(fact_246_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_247_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_248_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_249_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_250_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_251_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_252_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_253_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_254_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_255_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_256_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_257_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_258_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_259_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_260_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_261_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_262_int_Ozero__not__one,axiom,
    zero_zero_int != one_one_int ).

% int.zero_not_one
thf(fact_263_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_264_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_265_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_266_int_Oa__coset__add__assoc,axiom,
    ! [M3: set_int,G: int,H: int] :
      ( ( ord_less_eq_set_int @ M3 @ 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 ) ) ) @ M3 @ 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 ) ) ) @ M3 @ ( plus_plus_int @ G @ H ) ) ) ) ) ) ).

% int.a_coset_add_assoc
thf(fact_267_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_268_int_Oa__lcos__m__assoc,axiom,
    ! [M3: set_int,G: int,H: int] :
      ( ( ord_less_eq_set_int @ M3 @ 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 @ M3 ) )
            = ( 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 ) @ M3 ) ) ) ) ) ).

% int.a_lcos_m_assoc
thf(fact_269_int_Oa__lcos__mult__one,axiom,
    ! [M3: set_int] :
      ( ( ord_less_eq_set_int @ M3 @ 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 @ M3 )
        = M3 ) ) ).

% int.a_lcos_mult_one
thf(fact_270_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_271_int_Oa__coset__add__inv1,axiom,
    ! [M3: 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 ) ) ) @ M3 @ ( plus_plus_int @ X @ ( uminus_uminus_int @ Y ) ) )
        = M3 )
     => ( ( member_int @ X @ top_top_set_int )
       => ( ( member_int @ Y @ top_top_set_int )
         => ( ( ord_less_eq_set_int @ M3 @ 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 ) ) ) @ M3 @ 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 ) ) ) @ M3 @ Y ) ) ) ) ) ) ).

% int.a_coset_add_inv1
thf(fact_272_int_Oa__coset__add__inv2,axiom,
    ! [M3: 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 ) ) ) @ M3 @ 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 ) ) ) @ M3 @ Y ) )
     => ( ( member_int @ X @ top_top_set_int )
       => ( ( member_int @ Y @ top_top_set_int )
         => ( ( ord_less_eq_set_int @ M3 @ 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 ) ) ) @ M3 @ ( plus_plus_int @ X @ ( uminus_uminus_int @ Y ) ) )
              = M3 ) ) ) ) ) ).

% int.a_coset_add_inv2
thf(fact_273_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_274_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_275_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_276_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_277_int_Oadd_Ol__inv__ex,axiom,
    ! [X: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ? [X5: int] :
          ( ( member_int @ X5 @ top_top_set_int )
          & ( ( plus_plus_int @ X5 @ X )
            = zero_zero_int ) ) ) ).

% int.add.l_inv_ex
thf(fact_278_int_Oadd_Or__inv__ex,axiom,
    ! [X: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ? [X5: int] :
          ( ( member_int @ X5 @ top_top_set_int )
          & ( ( plus_plus_int @ X @ X5 )
            = zero_zero_int ) ) ) ).

% int.add.r_inv_ex
thf(fact_279_int_Oadd_Oinv__unique,axiom,
    ! [Y: int,X: int,Y3: int] :
      ( ( ( plus_plus_int @ Y @ X )
        = zero_zero_int )
     => ( ( ( plus_plus_int @ X @ Y3 )
          = zero_zero_int )
       => ( ( member_int @ X @ top_top_set_int )
         => ( ( member_int @ Y @ top_top_set_int )
           => ( ( member_int @ Y3 @ top_top_set_int )
             => ( Y = Y3 ) ) ) ) ) ) ).

% int.add.inv_unique
thf(fact_280_int_Oadd_Oone__unique,axiom,
    ! [U: int] :
      ( ( member_int @ U @ top_top_set_int )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ top_top_set_int )
           => ( ( plus_plus_int @ U @ X5 )
              = X5 ) )
       => ( U = zero_zero_int ) ) ) ).

% int.add.one_unique
thf(fact_281_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_282_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_283_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_284_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_285_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_286_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_287_int_Oinv__unique,axiom,
    ! [Y: int,X: int,Y3: int] :
      ( ( ( times_times_int @ Y @ X )
        = one_one_int )
     => ( ( ( times_times_int @ X @ Y3 )
          = one_one_int )
       => ( ( member_int @ X @ top_top_set_int )
         => ( ( member_int @ Y @ top_top_set_int )
           => ( ( member_int @ Y3 @ top_top_set_int )
             => ( Y = Y3 ) ) ) ) ) ) ).

% int.inv_unique
thf(fact_288_int_Oone__unique,axiom,
    ! [U: int] :
      ( ( member_int @ U @ top_top_set_int )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ top_top_set_int )
           => ( ( times_times_int @ U @ X5 )
              = X5 ) )
       => ( U = one_one_int ) ) ) ).

% int.one_unique
thf(fact_289_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_290_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_291_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_292_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_293_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_294_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_295_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_296_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_297_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_298_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_299_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_300_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_301_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_302_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_303_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_304_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_305_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_306_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_307_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_308_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_309_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_310_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_311_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 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ 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 ) ) ) @ X5 ) @ H2 ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ 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 ) ) ) @ X5 @ 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_312_i_Oa__rcosets__inv__mult__group__eq,axiom,
    ! [M3: set_int] :
      ( ( 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 ) ) ) @ ( 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 ) ) ) @ M3 ) @ M3 )
        = i ) ) ).

% i.a_rcosets_inv_mult_group_eq
thf(fact_313_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_314_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_315_negative__eq__positive,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
        = ( semiri1314217659103216013at_int @ M ) )
      = ( ( N2 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_316_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_317_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_318_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_319_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_320_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_321_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_322_int_Ole__refl,axiom,
    ! [X: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ord_less_eq_int @ X @ X ) ) ).

% int.le_refl
thf(fact_323_lessThan__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( set_ord_lessThan_nat @ X )
        = ( set_ord_lessThan_nat @ Y ) )
      = ( X = Y ) ) ).

% lessThan_eq_iff
thf(fact_324_finite__interval__int1,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_eq_int @ A @ I2 )
            & ( ord_less_eq_int @ I2 @ B ) ) ) ) ).

% finite_interval_int1
thf(fact_325_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_326_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_327_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_328_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_329_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_330_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_331_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_332_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_333_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_334_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_335_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_336_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_337_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_338_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_339_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_340_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_341_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_342_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_343_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_344_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_345_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_346_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_347_negative__zle,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_348_lessThan__iff,axiom,
    ! [I: set_set_int,K: set_set_int] :
      ( ( member_set_set_int @ I @ ( set_or4141871561713569845et_int @ K ) )
      = ( ord_less_set_set_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_349_lessThan__iff,axiom,
    ! [I: set_int,K: set_int] :
      ( ( member_set_int @ I @ ( set_or5935648273017318783et_int @ K ) )
      = ( ord_less_set_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_350_lessThan__iff,axiom,
    ! [I: int > set_int,K: int > set_int] :
      ( ( member_int_set_int @ I @ ( set_or7631710232683477194et_int @ K ) )
      = ( ord_less_int_set_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_351_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_352_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_353_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% nat_add_left_cancel_le
thf(fact_354_finite__interval__int2,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_eq_int @ A @ I2 )
            & ( ord_less_int @ I2 @ B ) ) ) ) ).

% finite_interval_int2
thf(fact_355_finite__interval__int3,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_int @ A @ I2 )
            & ( ord_less_eq_int @ I2 @ B ) ) ) ) ).

% finite_interval_int3
thf(fact_356_finite__interval__int4,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_int @ A @ I2 )
            & ( ord_less_int @ I2 @ B ) ) ) ) ).

% finite_interval_int4
thf(fact_357_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_358_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_359_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_360_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_361_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_362_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_363_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_364_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_365_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_366_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_367_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_368_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_neg_neg_trivial
thf(fact_369_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_pos_pos_trivial
thf(fact_370_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_371_i_Orcosets__add__eq,axiom,
    ! [M3: set_int] :
      ( ( 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 ) ) ) @ i @ M3 )
        = M3 ) ) ).

% i.rcosets_add_eq
thf(fact_372_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_373_i_Oa__rcosets__assoc,axiom,
    ! [M1: set_int,M22: set_int,M32: 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 @ M22 @ ( 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 @ M32 @ ( 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 @ M22 ) @ M32 )
            = ( 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 ) ) ) @ M22 @ M32 ) ) ) ) ) ) ).

% i.a_rcosets_assoc
thf(fact_374_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_375_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% mult_le_cancel2
thf(fact_376_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_377_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_378_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_379_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_380_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_381_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_382_int_Olless__eq,axiom,
    ( ord_less_int
    = ( ^ [X2: int,Y2: int] :
          ( ( ord_less_eq_int @ X2 @ Y2 )
          & ( X2 != Y2 ) ) ) ) ).

% int.lless_eq
thf(fact_383_zle__int,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% zle_int
thf(fact_384_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_385_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_386_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% eq_imp_le
thf(fact_387_le__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% le_antisym
thf(fact_388_nat__le__linear,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
      | ( ord_less_eq_nat @ N2 @ M ) ) ).

% nat_le_linear
thf(fact_389_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ? [X5: nat] :
            ( ( P @ X5 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_390_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M3: nat] :
      ( ( P @ X )
     => ( ! [X5: nat] :
            ( ( P @ X5 )
           => ( ord_less_eq_nat @ X5 @ M3 ) )
       => ~ ! [M4: nat] :
              ( ( P @ M4 )
             => ~ ! [X3: nat] :
                    ( ( P @ X3 )
                   => ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_391_infinite__UNIV__int,axiom,
    ~ ( finite_finite_int @ top_top_set_int ) ).

% infinite_UNIV_int
thf(fact_392_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_393_infinite__Iio,axiom,
    ! [A: int] :
      ~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).

% infinite_Iio
thf(fact_394_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_395_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_396_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_397_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_398_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_399_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_400_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N: nat] :
          ( ( ord_less_eq_nat @ M5 @ N )
          & ( M5 != N ) ) ) ) ).

% nat_less_le
thf(fact_401_less__imp__le__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_imp_le_nat
thf(fact_402_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N: nat] :
          ( ( ord_less_nat @ M5 @ N )
          | ( M5 = N ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_403_less__or__eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ord_less_nat @ M @ N2 )
        | ( M = N2 ) )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_404_le__neq__implies__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( M != N2 )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_405_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_406_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N: nat] :
        ? [K4: nat] :
          ( N
          = ( plus_plus_nat @ M5 @ K4 ) ) ) ) ).

% nat_le_iff_add
thf(fact_407_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_408_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_409_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_410_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_411_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N3: nat] :
          ( L
          = ( plus_plus_nat @ K @ N3 ) ) ) ).

% le_Suc_ex
thf(fact_412_add__leD2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ( ord_less_eq_nat @ K @ N2 ) ) ).

% add_leD2
thf(fact_413_add__leD1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% add_leD1
thf(fact_414_le__add2,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).

% le_add2
thf(fact_415_le__add1,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).

% le_add1
thf(fact_416_add__leE,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ~ ( ( ord_less_eq_nat @ M @ N2 )
         => ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% add_leE
thf(fact_417_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_418_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_419_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_420_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_421_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_422_mod__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ M ) ).

% mod_less_eq_dividend
thf(fact_423_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N3: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_424_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N3: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% nonneg_int_cases
thf(fact_425_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W2: int,Z2: int] :
        ? [N: nat] :
          ( Z2
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_426_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_427_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_428_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_429_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_430_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_431_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N3: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% nonpos_int_cases
thf(fact_432_negative__zle__0,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_433_int__zle__neg,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N2 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_434_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel1
thf(fact_435_ex__least__nat__le,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N2 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K3 )
               => ~ ( P @ I4 ) )
            & ( P @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_436_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_nat @ M4 @ N3 )
         => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_437_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).

% zmod_le_nonneg_dividend
thf(fact_438_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).

% neg_mod_sign
thf(fact_439_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).

% Euclidean_Division.pos_mod_sign
thf(fact_440_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( modulo_modulo_int @ I @ K )
        = I )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zmod_trivial_iff
thf(fact_441_int__int__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M = N2 ) ) ).

% int_int_eq
thf(fact_442_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_443_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_444_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( modulo_modulo_int @ K @ L )
          = ( plus_plus_int @ K @ L ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_445_mod__le__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).

% mod_le_divisor
thf(fact_446_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q3 )
        = ( modulo_modulo_nat @ N2 @ Q3 ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ~ ! [S2: nat] :
              ( N2
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q3 @ S2 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_447_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q3 )
        = ( modulo_modulo_nat @ N2 @ Q3 ) )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ~ ! [S2: nat] :
              ( M
             != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q3 @ S2 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_448_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R )
       => ( ( ord_less_int @ R @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R ) ) ) ) ).

% int_mod_pos_eq
thf(fact_449_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R ) )
     => ( ( ord_less_eq_int @ R @ zero_zero_int )
       => ( ( ord_less_int @ B @ R )
         => ( ( modulo_modulo_int @ A @ B )
            = R ) ) ) ) ).

% int_mod_neg_eq
thf(fact_450_split__zmod,axiom,
    ! [Q: int > $o,N2: int,K: int] :
      ( ( Q @ ( modulo_modulo_int @ N2 @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( Q @ N2 ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I2: int,J2: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
                & ( ord_less_int @ J2 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J2 ) ) )
             => ( Q @ J2 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I2: int,J2: int] :
              ( ( ( ord_less_int @ K @ J2 )
                & ( ord_less_eq_int @ J2 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J2 ) ) )
             => ( Q @ J2 ) ) ) ) ) ).

% split_zmod
thf(fact_451_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A @ C )
          = ( times_times_int @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_452_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C )
          = ( times_times_nat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_453_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A )
          = ( times_times_int @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_454_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A )
          = ( times_times_nat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_455_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( ( times_times_int @ A @ B )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_456_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ( times_times_nat @ A @ B )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_457_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_458_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_459_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
       != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_460_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
       != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_461_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_462_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_463_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_464_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_465_distrib__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% distrib_right
thf(fact_466_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% distrib_right
thf(fact_467_distrib__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% distrib_left
thf(fact_468_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% distrib_left
thf(fact_469_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_470_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_471_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_472_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_473_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus_int @ K @ zero_zero_int )
      = K ) ).

% plus_int_code(1)
thf(fact_474_plus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( plus_plus_int @ zero_zero_int @ L )
      = L ) ).

% plus_int_code(2)
thf(fact_475_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_476_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_477_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).

% zmult_zless_mono2
thf(fact_478_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_479_int__gr__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_480_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_481_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_482_minus__mult__commute,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).

% minus_mult_commute
thf(fact_483_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ A )
        = ( times_times_int @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_int @ B ) ) ) ) ).

% square_eq_iff
thf(fact_484_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N3: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N3 ) )
     => ~ ! [N3: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% int_cases2
thf(fact_485_not__int__zless__negative,axiom,
    ! [N2: nat,M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% not_int_zless_negative
thf(fact_486_lessThan__strict__subset__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N2 ) )
      = ( ord_less_int @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_487_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_488_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( K = zero_zero_nat )
        | ( M = N2 ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_489_lessThan__non__empty,axiom,
    ! [X: int] :
      ( ( set_ord_lessThan_int @ X )
     != bot_bot_set_int ) ).

% lessThan_non_empty
thf(fact_490_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_491_lambda__zero,axiom,
    ( ( ^ [H3: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_492_lambda__zero,axiom,
    ( ( ^ [H3: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_493_lambda__one,axiom,
    ( ( ^ [X2: int] : X2 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_494_lambda__one,axiom,
    ( ( ^ [X2: nat] : X2 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_495_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X2: int] : ( ord_less_int @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_496_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X2: nat] : ( ord_less_nat @ X2 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_497_int_Osetadd__subset__G,axiom,
    ! [H2: set_int,K2: set_int] :
      ( ( ord_less_eq_set_int @ H2 @ top_top_set_int )
     => ( ( ord_less_eq_set_int @ K2 @ 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 @ K2 ) @ top_top_set_int ) ) ) ).

% int.setadd_subset_G
thf(fact_498_int_Oset__add__closed,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ top_top_set_int )
     => ( ( ord_less_eq_set_int @ B3 @ 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 @ B3 ) @ top_top_set_int ) ) ) ).

% int.set_add_closed
thf(fact_499_int_Oset__add__comm,axiom,
    ! [I5: set_int,J4: set_int] :
      ( ( ord_less_eq_set_int @ I5 @ top_top_set_int )
     => ( ( ord_less_eq_set_int @ J4 @ 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 ) ) ) @ I5 @ J4 )
          = ( 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 ) ) ) @ J4 @ I5 ) ) ) ) ).

% int.set_add_comm
thf(fact_500_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_501_int_Oadd__additive__subgroups,axiom,
    ! [H2: set_int,K2: 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 @ K2 @ ( partia4118392927963588428t_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 @ K2 ) @ ( partia4118392927963588428t_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_502_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_503_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_504_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_505_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_506_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_507_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_508_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_509_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_510_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_511_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_512_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_513_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_514_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
          & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_515_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_516_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_517_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_518_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_519_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_520_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_521_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_522_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_523_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_524_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_525_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_526_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_527_mult__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_528_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_529_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_530_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_531_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_int @ zero_zero_int @ one_one_int ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_532_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_533_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_int @ zero_zero_int @ one_one_int ).

% zero_less_one_class.zero_le_one
thf(fact_534_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one_class.zero_le_one
thf(fact_535_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
     => ( ( ord_less_int @ X @ zero_zero_int )
        | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_536_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_537_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_538_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_539_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_540_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_541_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_542_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_543_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_544_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_545_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_546_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_547_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_548_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_549_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_550_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_551_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_552_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ zero_zero_int @ B ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_553_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_554_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_555_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_556_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_557_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_558_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_559_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_560_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_561_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ B @ zero_zero_int ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_562_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_563_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_564_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_565_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_566_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_567_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_568_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_569_add__mono1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).

% add_mono1
thf(fact_570_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_571_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_572_less__1__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_573_less__1__mult,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N2 )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_574_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_575_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_576_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N2 )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N2 = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_577_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( ( times_times_int @ X @ X )
        = one_one_int )
      = ( ( X = one_one_int )
        | ( X
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_578_int_Oa__setmult__rcos__assoc,axiom,
    ! [H2: set_int,K2: set_int,X: int] :
      ( ( ord_less_eq_set_int @ H2 @ top_top_set_int )
     => ( ( ord_less_eq_set_int @ K2 @ 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 ) ) ) @ K2 @ 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 @ K2 ) @ X ) ) ) ) ) ).

% int.a_setmult_rcos_assoc
thf(fact_579_zadd__int__left,axiom,
    ! [M: nat,N2: nat,Z: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) ) @ Z ) ) ).

% zadd_int_left
thf(fact_580_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N2: int] :
      ( ( ( times_times_int @ M @ N2 )
        = one_one_int )
     => ( ( M = one_one_int )
        | ( M
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_581_zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ( times_times_int @ M @ N2 )
        = one_one_int )
      = ( ( ( M = one_one_int )
          & ( N2 = one_one_int ) )
        | ( ( M
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N2
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_582_Iio__eq__empty__iff,axiom,
    ! [N2: nat] :
      ( ( ( set_ord_lessThan_nat @ N2 )
        = bot_bot_set_nat )
      = ( N2 = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
thf(fact_583_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel1
thf(fact_584_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N2 ) )
        = ( M = N2 ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_585_int_Oa__rcos__assoc__lcos,axiom,
    ! [H2: set_int,K2: set_int,X: int] :
      ( ( ord_less_eq_set_int @ H2 @ top_top_set_int )
     => ( ( ord_less_eq_set_int @ K2 @ 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 ) @ K2 )
            = ( 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 @ K2 ) ) ) ) ) ) ).

% int.a_rcos_assoc_lcos
thf(fact_586_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_587_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ C @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_588_mult__left__le,axiom,
    ! [C: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_589_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_590_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_591_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_592_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_593_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_594_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_595_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_596_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_597_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_598_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_599_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_600_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_601_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_602_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_603_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_604_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_605_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_606_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_607_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_608_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_609_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_610_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_611_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_612_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_613_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_614_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_615_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_616_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_617_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_618_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N3 )
          & ( K
            = ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_619_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N3: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N3 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% pos_int_cases
thf(fact_620_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_621_int__cases4,axiom,
    ! [M: int] :
      ( ! [N3: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N3 ) )
     => ~ ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% int_cases4
thf(fact_622_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ( ord_less_eq_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_623_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_624_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_625_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_626_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_627_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_628_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_629_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_630_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_631_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N3: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N3 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
       => ~ ! [N3: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).

% int_cases3
thf(fact_632_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N3: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% neg_int_cases
thf(fact_633_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_int @ X @ A )
     => ( ( ord_less_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_634_Sup__empty,axiom,
    ( ( comple3221217463730067765et_int @ bot_bot_set_set_int )
    = bot_bot_set_int ) ).

% Sup_empty
thf(fact_635_Sup__empty,axiom,
    ( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
    = bot_bot_set_nat ) ).

% Sup_empty
thf(fact_636_Sup__empty,axiom,
    ( ( complete_Sup_Sup_o @ bot_bot_set_o )
    = bot_bot_o ) ).

% Sup_empty
thf(fact_637_Sup__empty,axiom,
    ( ( comple7281953568134767595et_int @ bot_bo2384636101374064866et_int )
    = bot_bot_set_set_int ) ).

% Sup_empty
thf(fact_638_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_639_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_640_i_Oadd_Osurj__const__mult,axiom,
    ! [A: 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 ) ) ) ) )
     => ( ( image_int_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 ) @ ( 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 ) ) ) ) )
        = ( 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.surj_const_mult
thf(fact_641_boolean__algebra_Ocompl__one,axiom,
    ( ( uminus7346710233107665121et_int @ top_top_set_set_int )
    = bot_bot_set_set_int ) ).

% boolean_algebra.compl_one
thf(fact_642_boolean__algebra_Ocompl__one,axiom,
    ( ( uminus1532241313380277803et_int @ top_top_set_int )
    = bot_bot_set_int ) ).

% boolean_algebra.compl_one
thf(fact_643_boolean__algebra_Ocompl__one,axiom,
    ( ( uminus5710092332889474511et_nat @ top_top_set_nat )
    = bot_bot_set_nat ) ).

% boolean_algebra.compl_one
thf(fact_644_boolean__algebra_Ocompl__zero,axiom,
    ( ( uminus7346710233107665121et_int @ bot_bot_set_set_int )
    = top_top_set_set_int ) ).

% boolean_algebra.compl_zero
thf(fact_645_boolean__algebra_Ocompl__zero,axiom,
    ( ( uminus1532241313380277803et_int @ bot_bot_set_int )
    = top_top_set_int ) ).

% boolean_algebra.compl_zero
thf(fact_646_boolean__algebra_Ocompl__zero,axiom,
    ( ( uminus5710092332889474511et_nat @ bot_bot_set_nat )
    = top_top_set_nat ) ).

% boolean_algebra.compl_zero
thf(fact_647_Sup__UNIV,axiom,
    ( ( comple3221217463730067765et_int @ top_top_set_set_int )
    = top_top_set_int ) ).

% Sup_UNIV
thf(fact_648_Sup__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
    = top_top_set_nat ) ).

% Sup_UNIV
thf(fact_649_Sup__UNIV,axiom,
    ( ( complete_Sup_Sup_o @ top_top_set_o )
    = top_top_o ) ).

% Sup_UNIV
thf(fact_650_Sup__UNIV,axiom,
    ( ( comple7281953568134767595et_int @ top_to5524576366173240574et_int )
    = top_top_set_set_int ) ).

% Sup_UNIV
thf(fact_651_Union__iff,axiom,
    ! [A3: set_set_int,C2: set_set_set_set_int] :
      ( ( member_set_set_int @ A3 @ ( comple1756060948637632417et_int @ C2 ) )
      = ( ? [X2: set_set_set_int] :
            ( ( member7356822600254261989et_int @ X2 @ C2 )
            & ( member_set_set_int @ A3 @ X2 ) ) ) ) ).

% Union_iff
thf(fact_652_Union__iff,axiom,
    ! [A3: int > set_int,C2: set_set_int_set_int] :
      ( ( member_int_set_int @ A3 @ ( comple8192055500899288118et_int @ C2 ) )
      = ( ? [X2: set_int_set_int] :
            ( ( member7905727001773941114et_int @ X2 @ C2 )
            & ( member_int_set_int @ A3 @ X2 ) ) ) ) ).

% Union_iff
thf(fact_653_Union__iff,axiom,
    ! [A3: int,C2: set_set_int] :
      ( ( member_int @ A3 @ ( comple3221217463730067765et_int @ C2 ) )
      = ( ? [X2: set_int] :
            ( ( member_set_int @ X2 @ C2 )
            & ( member_int @ A3 @ X2 ) ) ) ) ).

% Union_iff
thf(fact_654_Union__iff,axiom,
    ! [A3: nat,C2: set_set_nat] :
      ( ( member_nat @ A3 @ ( comple7399068483239264473et_nat @ C2 ) )
      = ( ? [X2: set_nat] :
            ( ( member_set_nat @ X2 @ C2 )
            & ( member_nat @ A3 @ X2 ) ) ) ) ).

% Union_iff
thf(fact_655_Union__iff,axiom,
    ! [A3: set_int,C2: set_set_set_int] :
      ( ( member_set_int @ A3 @ ( comple7281953568134767595et_int @ C2 ) )
      = ( ? [X2: set_set_int] :
            ( ( member_set_set_int @ X2 @ C2 )
            & ( member_set_int @ A3 @ X2 ) ) ) ) ).

% Union_iff
thf(fact_656_UnionI,axiom,
    ! [X6: set_set_set_int,C2: set_set_set_set_int,A3: set_set_int] :
      ( ( member7356822600254261989et_int @ X6 @ C2 )
     => ( ( member_set_set_int @ A3 @ X6 )
       => ( member_set_set_int @ A3 @ ( comple1756060948637632417et_int @ C2 ) ) ) ) ).

% UnionI
thf(fact_657_UnionI,axiom,
    ! [X6: set_int_set_int,C2: set_set_int_set_int,A3: int > set_int] :
      ( ( member7905727001773941114et_int @ X6 @ C2 )
     => ( ( member_int_set_int @ A3 @ X6 )
       => ( member_int_set_int @ A3 @ ( comple8192055500899288118et_int @ C2 ) ) ) ) ).

% UnionI
thf(fact_658_UnionI,axiom,
    ! [X6: set_int,C2: set_set_int,A3: int] :
      ( ( member_set_int @ X6 @ C2 )
     => ( ( member_int @ A3 @ X6 )
       => ( member_int @ A3 @ ( comple3221217463730067765et_int @ C2 ) ) ) ) ).

% UnionI
thf(fact_659_UnionI,axiom,
    ! [X6: set_nat,C2: set_set_nat,A3: nat] :
      ( ( member_set_nat @ X6 @ C2 )
     => ( ( member_nat @ A3 @ X6 )
       => ( member_nat @ A3 @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ).

% UnionI
thf(fact_660_UnionI,axiom,
    ! [X6: set_set_int,C2: set_set_set_int,A3: set_int] :
      ( ( member_set_set_int @ X6 @ C2 )
     => ( ( member_set_int @ A3 @ X6 )
       => ( member_set_int @ A3 @ ( comple7281953568134767595et_int @ C2 ) ) ) ) ).

% UnionI
thf(fact_661_UN__ball__bex__simps_I1_J,axiom,
    ! [A3: set_set_int,P: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int @ X2 @ ( comple3221217463730067765et_int @ A3 ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: set_int] :
            ( ( member_set_int @ X2 @ A3 )
           => ! [Y2: int] :
                ( ( member_int @ Y2 @ X2 )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(1)
thf(fact_662_UN__ball__bex__simps_I1_J,axiom,
    ! [A3: set_set_nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ A3 ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A3 )
           => ! [Y2: nat] :
                ( ( member_nat @ Y2 @ X2 )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(1)
thf(fact_663_UN__ball__bex__simps_I1_J,axiom,
    ! [A3: set_set_set_int,P: set_int > $o] :
      ( ( ! [X2: set_int] :
            ( ( member_set_int @ X2 @ ( comple7281953568134767595et_int @ A3 ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: set_set_int] :
            ( ( member_set_set_int @ X2 @ A3 )
           => ! [Y2: set_int] :
                ( ( member_set_int @ Y2 @ X2 )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(1)
thf(fact_664_UN__ball__bex__simps_I3_J,axiom,
    ! [A3: set_set_int,P: int > $o] :
      ( ( ? [X2: int] :
            ( ( member_int @ X2 @ ( comple3221217463730067765et_int @ A3 ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: set_int] :
            ( ( member_set_int @ X2 @ A3 )
            & ? [Y2: int] :
                ( ( member_int @ Y2 @ X2 )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(3)
thf(fact_665_UN__ball__bex__simps_I3_J,axiom,
    ! [A3: set_set_nat,P: nat > $o] :
      ( ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ A3 ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A3 )
            & ? [Y2: nat] :
                ( ( member_nat @ Y2 @ X2 )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(3)
thf(fact_666_UN__ball__bex__simps_I3_J,axiom,
    ! [A3: set_set_set_int,P: set_int > $o] :
      ( ( ? [X2: set_int] :
            ( ( member_set_int @ X2 @ ( comple7281953568134767595et_int @ A3 ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: set_set_int] :
            ( ( member_set_set_int @ X2 @ A3 )
            & ? [Y2: set_int] :
                ( ( member_set_int @ Y2 @ X2 )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(3)
thf(fact_667_finite__lessThan,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).

% finite_lessThan
thf(fact_668_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_669_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_670_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_671_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_672_Sup__bot__conv_I1_J,axiom,
    ! [A3: set_set_int] :
      ( ( ( comple3221217463730067765et_int @ A3 )
        = bot_bot_set_int )
      = ( ! [X2: set_int] :
            ( ( member_set_int @ X2 @ A3 )
           => ( X2 = bot_bot_set_int ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_673_Sup__bot__conv_I1_J,axiom,
    ! [A3: set_set_nat] :
      ( ( ( comple7399068483239264473et_nat @ A3 )
        = bot_bot_set_nat )
      = ( ! [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A3 )
           => ( X2 = bot_bot_set_nat ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_674_Sup__bot__conv_I1_J,axiom,
    ! [A3: set_o] :
      ( ( ( complete_Sup_Sup_o @ A3 )
        = bot_bot_o )
      = ( ! [X2: $o] :
            ( ( member_o @ X2 @ A3 )
           => ( X2 = bot_bot_o ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_675_Sup__bot__conv_I1_J,axiom,
    ! [A3: set_set_set_int] :
      ( ( ( comple7281953568134767595et_int @ A3 )
        = bot_bot_set_set_int )
      = ( ! [X2: set_set_int] :
            ( ( member_set_set_int @ X2 @ A3 )
           => ( X2 = bot_bot_set_set_int ) ) ) ) ).

% Sup_bot_conv(1)
thf(fact_676_Sup__bot__conv_I2_J,axiom,
    ! [A3: set_set_int] :
      ( ( bot_bot_set_int
        = ( comple3221217463730067765et_int @ A3 ) )
      = ( ! [X2: set_int] :
            ( ( member_set_int @ X2 @ A3 )
           => ( X2 = bot_bot_set_int ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_677_Sup__bot__conv_I2_J,axiom,
    ! [A3: set_set_nat] :
      ( ( bot_bot_set_nat
        = ( comple7399068483239264473et_nat @ A3 ) )
      = ( ! [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A3 )
           => ( X2 = bot_bot_set_nat ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_678_Sup__bot__conv_I2_J,axiom,
    ! [A3: set_o] :
      ( ( bot_bot_o
        = ( complete_Sup_Sup_o @ A3 ) )
      = ( ! [X2: $o] :
            ( ( member_o @ X2 @ A3 )
           => ( X2 = bot_bot_o ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_679_Sup__bot__conv_I2_J,axiom,
    ! [A3: set_set_set_int] :
      ( ( bot_bot_set_set_int
        = ( comple7281953568134767595et_int @ A3 ) )
      = ( ! [X2: set_set_int] :
            ( ( member_set_set_int @ X2 @ A3 )
           => ( X2 = bot_bot_set_set_int ) ) ) ) ).

% Sup_bot_conv(2)
thf(fact_680_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_681_SUP__identity__eq,axiom,
    ! [A3: set_int] :
      ( ( complete_Sup_Sup_int
        @ ( image_int_int
          @ ^ [X2: int] : X2
          @ A3 ) )
      = ( complete_Sup_Sup_int @ A3 ) ) ).

% SUP_identity_eq
thf(fact_682_SUP__identity__eq,axiom,
    ! [A3: set_set_int] :
      ( ( comple3221217463730067765et_int
        @ ( image_524474410958335435et_int
          @ ^ [X2: set_int] : X2
          @ A3 ) )
      = ( comple3221217463730067765et_int @ A3 ) ) ).

% SUP_identity_eq
thf(fact_683_SUP__identity__eq,axiom,
    ! [A3: set_nat] :
      ( ( complete_Sup_Sup_nat
        @ ( image_nat_nat
          @ ^ [X2: nat] : X2
          @ A3 ) )
      = ( complete_Sup_Sup_nat @ A3 ) ) ).

% SUP_identity_eq
thf(fact_684_SUP__identity__eq,axiom,
    ! [A3: set_set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_7916887816326733075et_nat
          @ ^ [X2: set_nat] : X2
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ A3 ) ) ).

% SUP_identity_eq
thf(fact_685_SUP__identity__eq,axiom,
    ! [A3: set_o] :
      ( ( complete_Sup_Sup_o
        @ ( image_o_o
          @ ^ [X2: $o] : X2
          @ A3 ) )
      = ( complete_Sup_Sup_o @ A3 ) ) ).

% SUP_identity_eq
thf(fact_686_SUP__identity__eq,axiom,
    ! [A3: set_set_set_int] :
      ( ( comple7281953568134767595et_int
        @ ( image_2718908958957770551et_int
          @ ^ [X2: set_set_int] : X2
          @ A3 ) )
      = ( comple7281953568134767595et_int @ A3 ) ) ).

% SUP_identity_eq
thf(fact_687_SUP__bot__conv_I2_J,axiom,
    ! [B3: set_int > set_int,A3: set_set_int] :
      ( ( bot_bot_set_int
        = ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ B3 @ A3 ) ) )
      = ( ! [X2: set_int] :
            ( ( member_set_int @ X2 @ A3 )
           => ( ( B3 @ X2 )
              = bot_bot_set_int ) ) ) ) ).

% SUP_bot_conv(2)
thf(fact_688_SUP__bot__conv_I2_J,axiom,
    ! [B3: int > set_int,A3: set_int] :
      ( ( bot_bot_set_int
        = ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A3 ) ) )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ A3 )
           => ( ( B3 @ X2 )
              = bot_bot_set_int ) ) ) ) ).

% SUP_bot_conv(2)
thf(fact_689_SUP__bot__conv_I2_J,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ( bot_bot_set_nat
        = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
           => ( ( B3 @ X2 )
              = bot_bot_set_nat ) ) ) ) ).

% SUP_bot_conv(2)
thf(fact_690_SUP__bot__conv_I2_J,axiom,
    ! [B3: int > set_set_int,A3: set_int] :
      ( ( bot_bot_set_set_int
        = ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ B3 @ A3 ) ) )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ A3 )
           => ( ( B3 @ X2 )
              = bot_bot_set_set_int ) ) ) ) ).

% SUP_bot_conv(2)
thf(fact_691_SUP__bot__conv_I1_J,axiom,
    ! [B3: set_int > set_int,A3: set_set_int] :
      ( ( ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ B3 @ A3 ) )
        = bot_bot_set_int )
      = ( ! [X2: set_int] :
            ( ( member_set_int @ X2 @ A3 )
           => ( ( B3 @ X2 )
              = bot_bot_set_int ) ) ) ) ).

% SUP_bot_conv(1)
thf(fact_692_SUP__bot__conv_I1_J,axiom,
    ! [B3: int > set_int,A3: set_int] :
      ( ( ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A3 ) )
        = bot_bot_set_int )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ A3 )
           => ( ( B3 @ X2 )
              = bot_bot_set_int ) ) ) ) ).

% SUP_bot_conv(1)
thf(fact_693_SUP__bot__conv_I1_J,axiom,
    ! [B3: nat > set_nat,A3: set_nat] :
      ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) )
        = bot_bot_set_nat )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
           => ( ( B3 @ X2 )
              = bot_bot_set_nat ) ) ) ) ).

% SUP_bot_conv(1)
thf(fact_694_SUP__bot__conv_I1_J,axiom,
    ! [B3: int > set_set_int,A3: set_int] :
      ( ( ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ B3 @ A3 ) )
        = bot_bot_set_set_int )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ A3 )
           => ( ( B3 @ X2 )
              = bot_bot_set_set_int ) ) ) ) ).

% SUP_bot_conv(1)
thf(fact_695_SUP__bot,axiom,
    ! [A3: set_set_int] :
      ( ( comple3221217463730067765et_int
        @ ( image_524474410958335435et_int
          @ ^ [X2: set_int] : bot_bot_set_int
          @ A3 ) )
      = bot_bot_set_int ) ).

% SUP_bot
thf(fact_696_SUP__bot,axiom,
    ! [A3: set_int] :
      ( ( comple3221217463730067765et_int
        @ ( image_int_set_int
          @ ^ [X2: int] : bot_bot_set_int
          @ A3 ) )
      = bot_bot_set_int ) ).

% SUP_bot
thf(fact_697_SUP__bot,axiom,
    ! [A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X2: nat] : bot_bot_set_nat
          @ A3 ) )
      = bot_bot_set_nat ) ).

% SUP_bot
thf(fact_698_SUP__bot,axiom,
    ! [A3: set_int] :
      ( ( comple7281953568134767595et_int
        @ ( image_6617583118289273547et_int
          @ ^ [X2: int] : bot_bot_set_set_int
          @ A3 ) )
      = bot_bot_set_set_int ) ).

% SUP_bot
thf(fact_699_SUP__const,axiom,
    ! [A3: set_int,F: $o] :
      ( ( A3 != bot_bot_set_int )
     => ( ( complete_Sup_Sup_o
          @ ( image_int_o
            @ ^ [I2: int] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_700_SUP__const,axiom,
    ! [A3: set_nat,F: $o] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( complete_Sup_Sup_o
          @ ( image_nat_o
            @ ^ [I2: nat] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_701_SUP__const,axiom,
    ! [A3: set_int,F: set_int] :
      ( ( A3 != bot_bot_set_int )
     => ( ( comple3221217463730067765et_int
          @ ( image_int_set_int
            @ ^ [I2: int] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_702_SUP__const,axiom,
    ! [A3: set_nat,F: set_int] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( comple3221217463730067765et_int
          @ ( image_nat_set_int
            @ ^ [I2: nat] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_703_SUP__const,axiom,
    ! [A3: set_int,F: set_nat] :
      ( ( A3 != bot_bot_set_int )
     => ( ( comple7399068483239264473et_nat
          @ ( image_int_set_nat
            @ ^ [I2: int] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_704_SUP__const,axiom,
    ! [A3: set_nat,F: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( comple7399068483239264473et_nat
          @ ( image_nat_set_nat
            @ ^ [I2: nat] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_705_SUP__const,axiom,
    ! [A3: set_set_int,F: $o] :
      ( ( A3 != bot_bot_set_set_int )
     => ( ( complete_Sup_Sup_o
          @ ( image_set_int_o
            @ ^ [I2: set_int] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_706_SUP__const,axiom,
    ! [A3: set_set_int,F: set_int] :
      ( ( A3 != bot_bot_set_set_int )
     => ( ( comple3221217463730067765et_int
          @ ( image_524474410958335435et_int
            @ ^ [I2: set_int] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_707_SUP__const,axiom,
    ! [A3: set_set_int,F: set_nat] :
      ( ( A3 != bot_bot_set_set_int )
     => ( ( comple7399068483239264473et_nat
          @ ( image_4702325430467532143et_nat
            @ ^ [I2: set_int] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_708_SUP__const,axiom,
    ! [A3: set_int,F: set_set_int] :
      ( ( A3 != bot_bot_set_int )
     => ( ( comple7281953568134767595et_int
          @ ( image_6617583118289273547et_int
            @ ^ [I2: int] : F
            @ A3 ) )
        = F ) ) ).

% SUP_const
thf(fact_709_UN__extend__simps_I10_J,axiom,
    ! [B3: int > set_int,F: int > int,A3: set_int] :
      ( ( comple3221217463730067765et_int
        @ ( image_int_set_int
          @ ^ [A4: int] : ( B3 @ ( F @ A4 ) )
          @ A3 ) )
      = ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ ( image_int_int @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_710_UN__extend__simps_I10_J,axiom,
    ! [B3: int > set_nat,F: int > int,A3: set_int] :
      ( ( comple7399068483239264473et_nat
        @ ( image_int_set_nat
          @ ^ [A4: int] : ( B3 @ ( F @ A4 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B3 @ ( image_int_int @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_711_UN__extend__simps_I10_J,axiom,
    ! [B3: nat > set_nat,F: nat > nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [A4: nat] : ( B3 @ ( F @ A4 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( image_nat_nat @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_712_UN__extend__simps_I10_J,axiom,
    ! [B3: set_nat > set_int,F: nat > set_nat,A3: set_nat] :
      ( ( comple3221217463730067765et_int
        @ ( image_nat_set_int
          @ ^ [A4: nat] : ( B3 @ ( F @ A4 ) )
          @ A3 ) )
      = ( comple3221217463730067765et_int @ ( image_3739036796817536367et_int @ B3 @ ( image_nat_set_nat @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_713_UN__extend__simps_I10_J,axiom,
    ! [B3: int > set_int,F: set_int > int,A3: set_set_int] :
      ( ( comple3221217463730067765et_int
        @ ( image_524474410958335435et_int
          @ ^ [A4: set_int] : ( B3 @ ( F @ A4 ) )
          @ A3 ) )
      = ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ ( image_set_int_int @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_714_UN__extend__simps_I10_J,axiom,
    ! [B3: set_int > set_int,F: int > set_int,A3: set_int] :
      ( ( comple3221217463730067765et_int
        @ ( image_int_set_int
          @ ^ [A4: int] : ( B3 @ ( F @ A4 ) )
          @ A3 ) )
      = ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ B3 @ ( image_int_set_int @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_715_UN__extend__simps_I10_J,axiom,
    ! [B3: set_int > set_nat,F: int > set_int,A3: set_int] :
      ( ( comple7399068483239264473et_nat
        @ ( image_int_set_nat
          @ ^ [A4: int] : ( B3 @ ( F @ A4 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ B3 @ ( image_int_set_int @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_716_UN__extend__simps_I10_J,axiom,
    ! [B3: set_nat > set_nat,F: nat > set_nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [A4: nat] : ( B3 @ ( F @ A4 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B3 @ ( image_nat_set_nat @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_717_UN__extend__simps_I10_J,axiom,
    ! [B3: int > set_set_int,F: int > int,A3: set_int] :
      ( ( comple7281953568134767595et_int
        @ ( image_6617583118289273547et_int
          @ ^ [A4: int] : ( B3 @ ( F @ A4 ) )
          @ A3 ) )
      = ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ B3 @ ( image_int_int @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_718_UN__extend__simps_I10_J,axiom,
    ! [B3: set_int > set_int,F: set_int > set_int,A3: set_set_int] :
      ( ( comple3221217463730067765et_int
        @ ( image_524474410958335435et_int
          @ ^ [A4: set_int] : ( B3 @ ( F @ A4 ) )
          @ A3 ) )
      = ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ B3 @ ( image_524474410958335435et_int @ F @ A3 ) ) ) ) ).

% UN_extend_simps(10)
thf(fact_719_SUP__cong,axiom,
    ! [A3: set_int,B3: set_int,C2: int > int,D3: int > int] :
      ( ( A3 = B3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( complete_Sup_Sup_int @ ( image_int_int @ C2 @ A3 ) )
          = ( complete_Sup_Sup_int @ ( image_int_int @ D3 @ B3 ) ) ) ) ) ).

% SUP_cong
thf(fact_720_SUP__cong,axiom,
    ! [A3: set_nat,B3: set_nat,C2: nat > nat,D3: nat > nat] :
      ( ( A3 = B3 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C2 @ A3 ) )
          = ( complete_Sup_Sup_nat @ ( image_nat_nat @ D3 @ B3 ) ) ) ) ) ).

% SUP_cong
thf(fact_721_SUP__cong,axiom,
    ! [A3: set_int,B3: set_int,C2: int > nat,D3: int > nat] :
      ( ( A3 = B3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( complete_Sup_Sup_nat @ ( image_int_nat @ C2 @ A3 ) )
          = ( complete_Sup_Sup_nat @ ( image_int_nat @ D3 @ B3 ) ) ) ) ) ).

% SUP_cong
thf(fact_722_SUP__cong,axiom,
    ! [A3: set_nat,B3: set_nat,C2: nat > $o,D3: nat > $o] :
      ( ( A3 = B3 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_nat_o @ C2 @ A3 ) )
          = ( complete_Sup_Sup_o @ ( image_nat_o @ D3 @ B3 ) ) ) ) ) ).

% SUP_cong
thf(fact_723_SUP__cong,axiom,
    ! [A3: set_int,B3: set_int,C2: int > $o,D3: int > $o] :
      ( ( A3 = B3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_int_o @ C2 @ A3 ) )
          = ( complete_Sup_Sup_o @ ( image_int_o @ D3 @ B3 ) ) ) ) ) ).

% SUP_cong
thf(fact_724_SUP__cong,axiom,
    ! [A3: set_nat,B3: set_nat,C2: nat > set_int,D3: nat > set_int] :
      ( ( A3 = B3 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ C2 @ A3 ) )
          = ( comple3221217463730067765et_int @ ( image_nat_set_int @ D3 @ B3 ) ) ) ) ) ).

% SUP_cong
thf(fact_725_SUP__cong,axiom,
    ! [A3: set_int,B3: set_int,C2: int > set_int,D3: int > set_int] :
      ( ( A3 = B3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( comple3221217463730067765et_int @ ( image_int_set_int @ C2 @ A3 ) )
          = ( comple3221217463730067765et_int @ ( image_int_set_int @ D3 @ B3 ) ) ) ) ) ).

% SUP_cong
thf(fact_726_SUP__cong,axiom,
    ! [A3: set_set_int,B3: set_set_int,C2: set_int > nat,D3: set_int > nat] :
      ( ( A3 = B3 )
     => ( ! [X5: set_int] :
            ( ( member_set_int @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( complete_Sup_Sup_nat @ ( image_set_int_nat @ C2 @ A3 ) )
          = ( complete_Sup_Sup_nat @ ( image_set_int_nat @ D3 @ B3 ) ) ) ) ) ).

% SUP_cong
thf(fact_727_SUP__cong,axiom,
    ! [A3: set_nat,B3: set_nat,C2: nat > set_nat,D3: nat > set_nat] :
      ( ( A3 = B3 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ A3 ) )
          = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ D3 @ B3 ) ) ) ) ) ).

% SUP_cong
thf(fact_728_SUP__cong,axiom,
    ! [A3: set_int,B3: set_int,C2: int > set_nat,D3: int > set_nat] :
      ( ( A3 = B3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ C2 @ A3 ) )
          = ( comple7399068483239264473et_nat @ ( image_int_set_nat @ D3 @ B3 ) ) ) ) ) ).

% SUP_cong
thf(fact_729_SUP__commute,axiom,
    ! [F: set_int > set_int > set_int,B3: set_set_int,A3: set_set_int] :
      ( ( comple3221217463730067765et_int
        @ ( image_524474410958335435et_int
          @ ^ [I2: set_int] : ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ ( F @ I2 ) @ B3 ) )
          @ A3 ) )
      = ( comple3221217463730067765et_int
        @ ( image_524474410958335435et_int
          @ ^ [J2: set_int] :
              ( comple3221217463730067765et_int
              @ ( image_524474410958335435et_int
                @ ^ [I2: set_int] : ( F @ I2 @ J2 )
                @ A3 ) )
          @ B3 ) ) ) ).

% SUP_commute
thf(fact_730_SUP__commute,axiom,
    ! [F: set_int > int > set_int,B3: set_int,A3: set_set_int] :
      ( ( comple3221217463730067765et_int
        @ ( image_524474410958335435et_int
          @ ^ [I2: set_int] : ( comple3221217463730067765et_int @ ( image_int_set_int @ ( F @ I2 ) @ B3 ) )
          @ A3 ) )
      = ( comple3221217463730067765et_int
        @ ( image_int_set_int
          @ ^ [J2: int] :
              ( comple3221217463730067765et_int
              @ ( image_524474410958335435et_int
                @ ^ [I2: set_int] : ( F @ I2 @ J2 )
                @ A3 ) )
          @ B3 ) ) ) ).

% SUP_commute
thf(fact_731_SUP__commute,axiom,
    ! [F: int > set_int > set_int,B3: set_set_int,A3: set_int] :
      ( ( comple3221217463730067765et_int
        @ ( image_int_set_int
          @ ^ [I2: int] : ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ ( F @ I2 ) @ B3 ) )
          @ A3 ) )
      = ( comple3221217463730067765et_int
        @ ( image_524474410958335435et_int
          @ ^ [J2: set_int] :
              ( comple3221217463730067765et_int
              @ ( image_int_set_int
                @ ^ [I2: int] : ( F @ I2 @ J2 )
                @ A3 ) )
          @ B3 ) ) ) ).

% SUP_commute
thf(fact_732_SUP__commute,axiom,
    ! [F: int > int > set_int,B3: set_int,A3: set_int] :
      ( ( comple3221217463730067765et_int
        @ ( image_int_set_int
          @ ^ [I2: int] : ( comple3221217463730067765et_int @ ( image_int_set_int @ ( F @ I2 ) @ B3 ) )
          @ A3 ) )
      = ( comple3221217463730067765et_int
        @ ( image_int_set_int
          @ ^ [J2: int] :
              ( comple3221217463730067765et_int
              @ ( image_int_set_int
                @ ^ [I2: int] : ( F @ I2 @ J2 )
                @ A3 ) )
          @ B3 ) ) ) ).

% SUP_commute
thf(fact_733_SUP__commute,axiom,
    ! [F: nat > nat > set_nat,B3: set_nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [I2: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F @ I2 ) @ B3 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [J2: nat] :
              ( comple7399068483239264473et_nat
              @ ( image_nat_set_nat
                @ ^ [I2: nat] : ( F @ I2 @ J2 )
                @ A3 ) )
          @ B3 ) ) ) ).

% SUP_commute
thf(fact_734_SUP__commute,axiom,
    ! [F: int > int > set_set_int,B3: set_int,A3: set_int] :
      ( ( comple7281953568134767595et_int
        @ ( image_6617583118289273547et_int
          @ ^ [I2: int] : ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ ( F @ I2 ) @ B3 ) )
          @ A3 ) )
      = ( comple7281953568134767595et_int
        @ ( image_6617583118289273547et_int
          @ ^ [J2: int] :
              ( comple7281953568134767595et_int
              @ ( image_6617583118289273547et_int
                @ ^ [I2: int] : ( F @ I2 @ J2 )
                @ A3 ) )
          @ B3 ) ) ) ).

% SUP_commute
thf(fact_735_image__UN,axiom,
    ! [F: int > int,B3: int > set_int,A3: set_int] :
      ( ( image_int_int @ F @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A3 ) ) )
      = ( comple3221217463730067765et_int
        @ ( image_int_set_int
          @ ^ [X2: int] : ( image_int_int @ F @ ( B3 @ X2 ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_736_image__UN,axiom,
    ! [F: int > nat,B3: nat > set_int,A3: set_nat] :
      ( ( image_int_nat @ F @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B3 @ A3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X2: nat] : ( image_int_nat @ F @ ( B3 @ X2 ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_737_image__UN,axiom,
    ! [F: int > nat,B3: int > set_int,A3: set_int] :
      ( ( image_int_nat @ F @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_int_set_nat
          @ ^ [X2: int] : ( image_int_nat @ F @ ( B3 @ X2 ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_738_image__UN,axiom,
    ! [F: nat > int,B3: int > set_nat,A3: set_int] :
      ( ( image_nat_int @ F @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B3 @ A3 ) ) )
      = ( comple3221217463730067765et_int
        @ ( image_int_set_int
          @ ^ [X2: int] : ( image_nat_int @ F @ ( B3 @ X2 ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_739_image__UN,axiom,
    ! [F: nat > int,B3: nat > set_nat,A3: set_nat] :
      ( ( image_nat_int @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( comple3221217463730067765et_int
        @ ( image_nat_set_int
          @ ^ [X2: nat] : ( image_nat_int @ F @ ( B3 @ X2 ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_740_image__UN,axiom,
    ! [F: nat > nat,B3: nat > set_nat,A3: set_nat] :
      ( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [X2: nat] : ( image_nat_nat @ F @ ( B3 @ X2 ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_741_image__UN,axiom,
    ! [F: int > int,B3: set_int > set_int,A3: set_set_int] :
      ( ( image_int_int @ F @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ B3 @ A3 ) ) )
      = ( comple3221217463730067765et_int
        @ ( image_524474410958335435et_int
          @ ^ [X2: set_int] : ( image_int_int @ F @ ( B3 @ X2 ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_742_image__UN,axiom,
    ! [F: int > nat,B3: set_int > set_int,A3: set_set_int] :
      ( ( image_int_nat @ F @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ B3 @ A3 ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_4702325430467532143et_nat
          @ ^ [X2: set_int] : ( image_int_nat @ F @ ( B3 @ X2 ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_743_image__UN,axiom,
    ! [F: int > set_int,B3: int > set_int,A3: set_int] :
      ( ( image_int_set_int @ F @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A3 ) ) )
      = ( comple7281953568134767595et_int
        @ ( image_6617583118289273547et_int
          @ ^ [X2: int] : ( image_int_set_int @ F @ ( B3 @ X2 ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_744_image__UN,axiom,
    ! [F: nat > set_nat,B3: nat > set_nat,A3: set_nat] :
      ( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( comple548664676211718543et_nat
        @ ( image_2194112158459175443et_nat
          @ ^ [X2: nat] : ( image_nat_set_nat @ F @ ( B3 @ X2 ) )
          @ A3 ) ) ) ).

% image_UN
thf(fact_745_SUP__UNION,axiom,
    ! [F: int > $o,G: int > set_int,A3: set_int] :
      ( ( complete_Sup_Sup_o @ ( image_int_o @ F @ ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ A3 ) ) ) )
      = ( complete_Sup_Sup_o
        @ ( image_int_o
          @ ^ [Y2: int] : ( complete_Sup_Sup_o @ ( image_int_o @ F @ ( G @ Y2 ) ) )
          @ A3 ) ) ) ).

% SUP_UNION
thf(fact_746_SUP__UNION,axiom,
    ! [F: nat > $o,G: nat > set_nat,A3: set_nat] :
      ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A3 ) ) ) )
      = ( complete_Sup_Sup_o
        @ ( image_nat_o
          @ ^ [Y2: nat] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( G @ Y2 ) ) )
          @ A3 ) ) ) ).

% SUP_UNION
thf(fact_747_SUP__UNION,axiom,
    ! [F: int > set_int,G: int > set_int,A3: set_int] :
      ( ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ A3 ) ) ) )
      = ( comple3221217463730067765et_int
        @ ( image_int_set_int
          @ ^ [Y2: int] : ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ ( G @ Y2 ) ) )
          @ A3 ) ) ) ).

% SUP_UNION
thf(fact_748_SUP__UNION,axiom,
    ! [F: nat > set_int,G: int > set_nat,A3: set_int] :
      ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ G @ A3 ) ) ) )
      = ( comple3221217463730067765et_int
        @ ( image_int_set_int
          @ ^ [Y2: int] : ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ ( G @ Y2 ) ) )
          @ A3 ) ) ) ).

% SUP_UNION
thf(fact_749_SUP__UNION,axiom,
    ! [F: nat > set_int,G: nat > set_nat,A3: set_nat] :
      ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A3 ) ) ) )
      = ( comple3221217463730067765et_int
        @ ( image_nat_set_int
          @ ^ [Y2: nat] : ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ ( G @ Y2 ) ) )
          @ A3 ) ) ) ).

% SUP_UNION
thf(fact_750_SUP__UNION,axiom,
    ! [F: int > set_nat,G: nat > set_int,A3: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ G @ A3 ) ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [Y2: nat] : ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ ( G @ Y2 ) ) )
          @ A3 ) ) ) ).

% SUP_UNION
thf(fact_751_SUP__UNION,axiom,
    ! [F: int > set_nat,G: int > set_int,A3: set_int] :
      ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ A3 ) ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_int_set_nat
          @ ^ [Y2: int] : ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ ( G @ Y2 ) ) )
          @ A3 ) ) ) ).

% SUP_UNION
thf(fact_752_SUP__UNION,axiom,
    ! [F: nat > set_nat,G: nat > set_nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A3 ) ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [Y2: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( G @ Y2 ) ) )
          @ A3 ) ) ) ).

% SUP_UNION
thf(fact_753_SUP__UNION,axiom,
    ! [F: int > $o,G: set_int > set_int,A3: set_set_int] :
      ( ( complete_Sup_Sup_o @ ( image_int_o @ F @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ G @ A3 ) ) ) )
      = ( complete_Sup_Sup_o
        @ ( image_set_int_o
          @ ^ [Y2: set_int] : ( complete_Sup_Sup_o @ ( image_int_o @ F @ ( G @ Y2 ) ) )
          @ A3 ) ) ) ).

% SUP_UNION
thf(fact_754_SUP__UNION,axiom,
    ! [F: set_int > $o,G: int > set_set_int,A3: set_int] :
      ( ( complete_Sup_Sup_o @ ( image_set_int_o @ F @ ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ G @ A3 ) ) ) )
      = ( complete_Sup_Sup_o
        @ ( image_int_o
          @ ^ [Y2: int] : ( complete_Sup_Sup_o @ ( image_set_int_o @ F @ ( G @ Y2 ) ) )
          @ A3 ) ) ) ).

% SUP_UNION
thf(fact_755_image__Union,axiom,
    ! [F: int > int,S3: set_set_int] :
      ( ( image_int_int @ F @ ( comple3221217463730067765et_int @ S3 ) )
      = ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ ( image_int_int @ F ) @ S3 ) ) ) ).

% image_Union
thf(fact_756_image__Union,axiom,
    ! [F: int > nat,S3: set_set_int] :
      ( ( image_int_nat @ F @ ( comple3221217463730067765et_int @ S3 ) )
      = ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ ( image_int_nat @ F ) @ S3 ) ) ) ).

% image_Union
thf(fact_757_image__Union,axiom,
    ! [F: nat > int,S3: set_set_nat] :
      ( ( image_nat_int @ F @ ( comple7399068483239264473et_nat @ S3 ) )
      = ( comple3221217463730067765et_int @ ( image_3739036796817536367et_int @ ( image_nat_int @ F ) @ S3 ) ) ) ).

% image_Union
thf(fact_758_image__Union,axiom,
    ! [F: nat > nat,S3: set_set_nat] :
      ( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ S3 ) )
      = ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ S3 ) ) ) ).

% image_Union
thf(fact_759_image__Union,axiom,
    ! [F: int > set_int,S3: set_set_int] :
      ( ( image_int_set_int @ F @ ( comple3221217463730067765et_int @ S3 ) )
      = ( comple7281953568134767595et_int @ ( image_1010086626112315521et_int @ ( image_int_set_int @ F ) @ S3 ) ) ) ).

% image_Union
thf(fact_760_image__Union,axiom,
    ! [F: nat > set_nat,S3: set_set_nat] :
      ( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ S3 ) )
      = ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ S3 ) ) ) ).

% image_Union
thf(fact_761_image__Union,axiom,
    ! [F: nat > set_int,S3: set_set_nat] :
      ( ( image_nat_set_int @ F @ ( comple7399068483239264473et_nat @ S3 ) )
      = ( comple7281953568134767595et_int @ ( image_4234937972324292645et_int @ ( image_nat_set_int @ F ) @ S3 ) ) ) ).

% image_Union
thf(fact_762_image__Union,axiom,
    ! [F: set_int > int,S3: set_set_set_int] :
      ( ( image_set_int_int @ F @ ( comple7281953568134767595et_int @ S3 ) )
      = ( comple3221217463730067765et_int @ ( image_3513010637850279041et_int @ ( image_set_int_int @ F ) @ S3 ) ) ) ).

% image_Union
thf(fact_763_image__Union,axiom,
    ! [F: set_int > nat,S3: set_set_set_int] :
      ( ( image_set_int_nat @ F @ ( comple7281953568134767595et_int @ S3 ) )
      = ( comple7399068483239264473et_nat @ ( image_7690861657359475749et_nat @ ( image_set_int_nat @ F ) @ S3 ) ) ) ).

% image_Union
thf(fact_764_image__Union,axiom,
    ! [F: int > set_set_int,S3: set_set_int] :
      ( ( image_6617583118289273547et_int @ F @ ( comple3221217463730067765et_int @ S3 ) )
      = ( comple1756060948637632417et_int @ ( image_1737623724905622583et_int @ ( image_6617583118289273547et_int @ F ) @ S3 ) ) ) ).

% image_Union
thf(fact_765_Sup_OSUP__identity__eq,axiom,
    ! [Sup: set_set_int > set_int,A3: set_set_int] :
      ( ( Sup
        @ ( image_524474410958335435et_int
          @ ^ [X2: set_int] : X2
          @ A3 ) )
      = ( Sup @ A3 ) ) ).

% Sup.SUP_identity_eq
thf(fact_766_Sup_OSUP__identity__eq,axiom,
    ! [Sup: set_int > int,A3: set_int] :
      ( ( Sup
        @ ( image_int_int
          @ ^ [X2: int] : X2
          @ A3 ) )
      = ( Sup @ A3 ) ) ).

% Sup.SUP_identity_eq
thf(fact_767_Inf_OINF__identity__eq,axiom,
    ! [Inf: set_set_int > set_int,A3: set_set_int] :
      ( ( Inf
        @ ( image_524474410958335435et_int
          @ ^ [X2: set_int] : X2
          @ A3 ) )
      = ( Inf @ A3 ) ) ).

% Inf.INF_identity_eq
thf(fact_768_Inf_OINF__identity__eq,axiom,
    ! [Inf: set_int > int,A3: set_int] :
      ( ( Inf
        @ ( image_int_int
          @ ^ [X2: int] : X2
          @ A3 ) )
      = ( Inf @ A3 ) ) ).

% Inf.INF_identity_eq
thf(fact_769_Sup_OSUP__cong,axiom,
    ! [A3: set_set_int,B3: set_set_int,C2: set_int > set_int,D3: set_int > set_int,Sup: set_set_int > set_int] :
      ( ( A3 = B3 )
     => ( ! [X5: set_int] :
            ( ( member_set_int @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( Sup @ ( image_524474410958335435et_int @ C2 @ A3 ) )
          = ( Sup @ ( image_524474410958335435et_int @ D3 @ B3 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_770_Sup_OSUP__cong,axiom,
    ! [A3: set_nat,B3: set_nat,C2: nat > set_nat,D3: nat > set_nat,Sup: set_set_nat > set_nat] :
      ( ( A3 = B3 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( Sup @ ( image_nat_set_nat @ C2 @ A3 ) )
          = ( Sup @ ( image_nat_set_nat @ D3 @ B3 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_771_Sup_OSUP__cong,axiom,
    ! [A3: set_int,B3: set_int,C2: int > set_set_int,D3: int > set_set_int,Sup: set_set_set_int > set_set_int] :
      ( ( A3 = B3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( Sup @ ( image_6617583118289273547et_int @ C2 @ A3 ) )
          = ( Sup @ ( image_6617583118289273547et_int @ D3 @ B3 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_772_Sup_OSUP__cong,axiom,
    ! [A3: set_int,B3: set_int,C2: int > set_int,D3: int > set_int,Sup: set_set_int > set_int] :
      ( ( A3 = B3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( Sup @ ( image_int_set_int @ C2 @ A3 ) )
          = ( Sup @ ( image_int_set_int @ D3 @ B3 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_773_Sup_OSUP__cong,axiom,
    ! [A3: set_int,B3: set_int,C2: int > int,D3: int > int,Sup: set_int > int] :
      ( ( A3 = B3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( Sup @ ( image_int_int @ C2 @ A3 ) )
          = ( Sup @ ( image_int_int @ D3 @ B3 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_774_Inf_OINF__cong,axiom,
    ! [A3: set_set_int,B3: set_set_int,C2: set_int > set_int,D3: set_int > set_int,Inf: set_set_int > set_int] :
      ( ( A3 = B3 )
     => ( ! [X5: set_int] :
            ( ( member_set_int @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( Inf @ ( image_524474410958335435et_int @ C2 @ A3 ) )
          = ( Inf @ ( image_524474410958335435et_int @ D3 @ B3 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_775_Inf_OINF__cong,axiom,
    ! [A3: set_nat,B3: set_nat,C2: nat > set_nat,D3: nat > set_nat,Inf: set_set_nat > set_nat] :
      ( ( A3 = B3 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( Inf @ ( image_nat_set_nat @ C2 @ A3 ) )
          = ( Inf @ ( image_nat_set_nat @ D3 @ B3 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_776_Inf_OINF__cong,axiom,
    ! [A3: set_int,B3: set_int,C2: int > set_set_int,D3: int > set_set_int,Inf: set_set_set_int > set_set_int] :
      ( ( A3 = B3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( Inf @ ( image_6617583118289273547et_int @ C2 @ A3 ) )
          = ( Inf @ ( image_6617583118289273547et_int @ D3 @ B3 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_777_Inf_OINF__cong,axiom,
    ! [A3: set_int,B3: set_int,C2: int > set_int,D3: int > set_int,Inf: set_set_int > set_int] :
      ( ( A3 = B3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( Inf @ ( image_int_set_int @ C2 @ A3 ) )
          = ( Inf @ ( image_int_set_int @ D3 @ B3 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_778_Inf_OINF__cong,axiom,
    ! [A3: set_int,B3: set_int,C2: int > int,D3: int > int,Inf: set_int > int] :
      ( ( A3 = B3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ B3 )
           => ( ( C2 @ X5 )
              = ( D3 @ X5 ) ) )
       => ( ( Inf @ ( image_int_int @ C2 @ A3 ) )
          = ( Inf @ ( image_int_int @ D3 @ B3 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_779_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_780_bounded__nat__set__is__finite,axiom,
    ! [N4: set_nat,N2: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ N4 )
         => ( ord_less_nat @ X5 @ N2 ) )
     => ( finite_finite_nat @ N4 ) ) ).

% bounded_nat_set_is_finite
thf(fact_781_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M5: nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ N5 )
         => ( ord_less_nat @ X2 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_782_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M5: nat] :
        ! [X2: nat] :
          ( ( member_nat @ X2 @ N5 )
         => ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_783_SUP__eq,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > $o,G: nat > $o] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X3 ) ) ) )
     => ( ! [J3: nat] :
            ( ( member_nat @ J3 @ B3 )
           => ? [X3: nat] :
                ( ( member_nat @ X3 @ A3 )
                & ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A3 ) )
          = ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_784_SUP__eq,axiom,
    ! [A3: set_nat,B3: set_int,F: nat > $o,G: int > $o] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X3 ) ) ) )
     => ( ! [J3: int] :
            ( ( member_int @ J3 @ B3 )
           => ? [X3: nat] :
                ( ( member_nat @ X3 @ A3 )
                & ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A3 ) )
          = ( complete_Sup_Sup_o @ ( image_int_o @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_785_SUP__eq,axiom,
    ! [A3: set_int,B3: set_nat,F: int > $o,G: nat > $o] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X3 ) ) ) )
     => ( ! [J3: nat] :
            ( ( member_nat @ J3 @ B3 )
           => ? [X3: int] :
                ( ( member_int @ X3 @ A3 )
                & ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_int_o @ F @ A3 ) )
          = ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_786_SUP__eq,axiom,
    ! [A3: set_int,B3: set_int,F: int > $o,G: int > $o] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ( ord_less_eq_o @ ( F @ I3 ) @ ( G @ X3 ) ) ) )
     => ( ! [J3: int] :
            ( ( member_int @ J3 @ B3 )
           => ? [X3: int] :
                ( ( member_int @ X3 @ A3 )
                & ( ord_less_eq_o @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
       => ( ( complete_Sup_Sup_o @ ( image_int_o @ F @ A3 ) )
          = ( complete_Sup_Sup_o @ ( image_int_o @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_787_SUP__eq,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > set_int,G: nat > set_int] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ( ord_less_eq_set_int @ ( F @ I3 ) @ ( G @ X3 ) ) ) )
     => ( ! [J3: nat] :
            ( ( member_nat @ J3 @ B3 )
           => ? [X3: nat] :
                ( ( member_nat @ X3 @ A3 )
                & ( ord_less_eq_set_int @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
       => ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) )
          = ( comple3221217463730067765et_int @ ( image_nat_set_int @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_788_SUP__eq,axiom,
    ! [A3: set_nat,B3: set_int,F: nat > set_int,G: int > set_int] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ( ord_less_eq_set_int @ ( F @ I3 ) @ ( G @ X3 ) ) ) )
     => ( ! [J3: int] :
            ( ( member_int @ J3 @ B3 )
           => ? [X3: nat] :
                ( ( member_nat @ X3 @ A3 )
                & ( ord_less_eq_set_int @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
       => ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) )
          = ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_789_SUP__eq,axiom,
    ! [A3: set_int,B3: set_nat,F: int > set_int,G: nat > set_int] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ( ord_less_eq_set_int @ ( F @ I3 ) @ ( G @ X3 ) ) ) )
     => ( ! [J3: nat] :
            ( ( member_nat @ J3 @ B3 )
           => ? [X3: int] :
                ( ( member_int @ X3 @ A3 )
                & ( ord_less_eq_set_int @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
       => ( ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) )
          = ( comple3221217463730067765et_int @ ( image_nat_set_int @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_790_SUP__eq,axiom,
    ! [A3: set_int,B3: set_int,F: int > set_int,G: int > set_int] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ( ord_less_eq_set_int @ ( F @ I3 ) @ ( G @ X3 ) ) ) )
     => ( ! [J3: int] :
            ( ( member_int @ J3 @ B3 )
           => ? [X3: int] :
                ( ( member_int @ X3 @ A3 )
                & ( ord_less_eq_set_int @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
       => ( ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) )
          = ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_791_SUP__eq,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > set_nat,G: nat > set_nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X3 ) ) ) )
     => ( ! [J3: nat] :
            ( ( member_nat @ J3 @ B3 )
           => ? [X3: nat] :
                ( ( member_nat @ X3 @ A3 )
                & ( ord_less_eq_set_nat @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
       => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) )
          = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_792_SUP__eq,axiom,
    ! [A3: set_nat,B3: set_int,F: nat > set_nat,G: int > set_nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ( ord_less_eq_set_nat @ ( F @ I3 ) @ ( G @ X3 ) ) ) )
     => ( ! [J3: int] :
            ( ( member_int @ J3 @ B3 )
           => ? [X3: nat] :
                ( ( member_nat @ X3 @ A3 )
                & ( ord_less_eq_set_nat @ ( G @ J3 ) @ ( F @ X3 ) ) ) )
       => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) )
          = ( comple7399068483239264473et_nat @ ( image_int_set_nat @ G @ B3 ) ) ) ) ) ).

% SUP_eq
thf(fact_793_SUP__eq__const,axiom,
    ! [I5: set_int,F: int > $o,X: $o] :
      ( ( I5 != bot_bot_set_int )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( complete_Sup_Sup_o @ ( image_int_o @ F @ I5 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_794_SUP__eq__const,axiom,
    ! [I5: set_nat,F: nat > $o,X: $o] :
      ( ( I5 != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I5 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_795_SUP__eq__const,axiom,
    ! [I5: set_int,F: int > set_int,X: set_int] :
      ( ( I5 != bot_bot_set_int )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ I5 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_796_SUP__eq__const,axiom,
    ! [I5: set_nat,F: nat > set_int,X: set_int] :
      ( ( I5 != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ I5 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_797_SUP__eq__const,axiom,
    ! [I5: set_int,F: int > set_nat,X: set_nat] :
      ( ( I5 != bot_bot_set_int )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ I5 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_798_SUP__eq__const,axiom,
    ! [I5: set_nat,F: nat > set_nat,X: set_nat] :
      ( ( I5 != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I5 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_799_SUP__eq__const,axiom,
    ! [I5: set_set_int,F: set_int > $o,X: $o] :
      ( ( I5 != bot_bot_set_set_int )
     => ( ! [I3: set_int] :
            ( ( member_set_int @ I3 @ I5 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( complete_Sup_Sup_o @ ( image_set_int_o @ F @ I5 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_800_SUP__eq__const,axiom,
    ! [I5: set_set_int,F: set_int > set_int,X: set_int] :
      ( ( I5 != bot_bot_set_set_int )
     => ( ! [I3: set_int] :
            ( ( member_set_int @ I3 @ I5 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ I5 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_801_SUP__eq__const,axiom,
    ! [I5: set_set_int,F: set_int > set_nat,X: set_nat] :
      ( ( I5 != bot_bot_set_set_int )
     => ( ! [I3: set_int] :
            ( ( member_set_int @ I3 @ I5 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ F @ I5 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_802_SUP__eq__const,axiom,
    ! [I5: set_set_set_int,F: set_set_int > $o,X: $o] :
      ( ( I5 != bot_bo2384636101374064866et_int )
     => ( ! [I3: set_set_int] :
            ( ( member_set_set_int @ I3 @ I5 )
           => ( ( F @ I3 )
              = X ) )
       => ( ( complete_Sup_Sup_o @ ( image_set_set_int_o @ F @ I5 ) )
          = X ) ) ) ).

% SUP_eq_const
thf(fact_803_SUP__eqI,axiom,
    ! [A3: set_nat,F: nat > $o,X: $o] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_less_eq_o @ ( F @ I3 ) @ X ) )
     => ( ! [Y4: $o] :
            ( ! [I4: nat] :
                ( ( member_nat @ I4 @ A3 )
               => ( ord_less_eq_o @ ( F @ I4 ) @ Y4 ) )
           => ( ord_less_eq_o @ X @ Y4 ) )
       => ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A3 ) )
          = X ) ) ) ).

% SUP_eqI
thf(fact_804_SUP__eqI,axiom,
    ! [A3: set_int,F: int > $o,X: $o] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ( ord_less_eq_o @ ( F @ I3 ) @ X ) )
     => ( ! [Y4: $o] :
            ( ! [I4: int] :
                ( ( member_int @ I4 @ A3 )
               => ( ord_less_eq_o @ ( F @ I4 ) @ Y4 ) )
           => ( ord_less_eq_o @ X @ Y4 ) )
       => ( ( complete_Sup_Sup_o @ ( image_int_o @ F @ A3 ) )
          = X ) ) ) ).

% SUP_eqI
thf(fact_805_SUP__eqI,axiom,
    ! [A3: set_nat,F: nat > set_int,X: set_int] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_less_eq_set_int @ ( F @ I3 ) @ X ) )
     => ( ! [Y4: set_int] :
            ( ! [I4: nat] :
                ( ( member_nat @ I4 @ A3 )
               => ( ord_less_eq_set_int @ ( F @ I4 ) @ Y4 ) )
           => ( ord_less_eq_set_int @ X @ Y4 ) )
       => ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) )
          = X ) ) ) ).

% SUP_eqI
thf(fact_806_SUP__eqI,axiom,
    ! [A3: set_int,F: int > set_int,X: set_int] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ( ord_less_eq_set_int @ ( F @ I3 ) @ X ) )
     => ( ! [Y4: set_int] :
            ( ! [I4: int] :
                ( ( member_int @ I4 @ A3 )
               => ( ord_less_eq_set_int @ ( F @ I4 ) @ Y4 ) )
           => ( ord_less_eq_set_int @ X @ Y4 ) )
       => ( ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) )
          = X ) ) ) ).

% SUP_eqI
thf(fact_807_SUP__eqI,axiom,
    ! [A3: set_nat,F: nat > set_nat,X: set_nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_less_eq_set_nat @ ( F @ I3 ) @ X ) )
     => ( ! [Y4: set_nat] :
            ( ! [I4: nat] :
                ( ( member_nat @ I4 @ A3 )
               => ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y4 ) )
           => ( ord_less_eq_set_nat @ X @ Y4 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) )
          = X ) ) ) ).

% SUP_eqI
thf(fact_808_SUP__eqI,axiom,
    ! [A3: set_int,F: int > set_nat,X: set_nat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ( ord_less_eq_set_nat @ ( F @ I3 ) @ X ) )
     => ( ! [Y4: set_nat] :
            ( ! [I4: int] :
                ( ( member_int @ I4 @ A3 )
               => ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y4 ) )
           => ( ord_less_eq_set_nat @ X @ Y4 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) )
          = X ) ) ) ).

% SUP_eqI
thf(fact_809_SUP__eqI,axiom,
    ! [A3: set_set_int,F: set_int > $o,X: $o] :
      ( ! [I3: set_int] :
          ( ( member_set_int @ I3 @ A3 )
         => ( ord_less_eq_o @ ( F @ I3 ) @ X ) )
     => ( ! [Y4: $o] :
            ( ! [I4: set_int] :
                ( ( member_set_int @ I4 @ A3 )
               => ( ord_less_eq_o @ ( F @ I4 ) @ Y4 ) )
           => ( ord_less_eq_o @ X @ Y4 ) )
       => ( ( complete_Sup_Sup_o @ ( image_set_int_o @ F @ A3 ) )
          = X ) ) ) ).

% SUP_eqI
thf(fact_810_SUP__eqI,axiom,
    ! [A3: set_set_int,F: set_int > set_int,X: set_int] :
      ( ! [I3: set_int] :
          ( ( member_set_int @ I3 @ A3 )
         => ( ord_less_eq_set_int @ ( F @ I3 ) @ X ) )
     => ( ! [Y4: set_int] :
            ( ! [I4: set_int] :
                ( ( member_set_int @ I4 @ A3 )
               => ( ord_less_eq_set_int @ ( F @ I4 ) @ Y4 ) )
           => ( ord_less_eq_set_int @ X @ Y4 ) )
       => ( ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ A3 ) )
          = X ) ) ) ).

% SUP_eqI
thf(fact_811_SUP__eqI,axiom,
    ! [A3: set_set_int,F: set_int > set_nat,X: set_nat] :
      ( ! [I3: set_int] :
          ( ( member_set_int @ I3 @ A3 )
         => ( ord_less_eq_set_nat @ ( F @ I3 ) @ X ) )
     => ( ! [Y4: set_nat] :
            ( ! [I4: set_int] :
                ( ( member_set_int @ I4 @ A3 )
               => ( ord_less_eq_set_nat @ ( F @ I4 ) @ Y4 ) )
           => ( ord_less_eq_set_nat @ X @ Y4 ) )
       => ( ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ F @ A3 ) )
          = X ) ) ) ).

% SUP_eqI
thf(fact_812_SUP__eqI,axiom,
    ! [A3: set_set_set_int,F: set_set_int > $o,X: $o] :
      ( ! [I3: set_set_int] :
          ( ( member_set_set_int @ I3 @ A3 )
         => ( ord_less_eq_o @ ( F @ I3 ) @ X ) )
     => ( ! [Y4: $o] :
            ( ! [I4: set_set_int] :
                ( ( member_set_set_int @ I4 @ A3 )
               => ( ord_less_eq_o @ ( F @ I4 ) @ Y4 ) )
           => ( ord_less_eq_o @ X @ Y4 ) )
       => ( ( complete_Sup_Sup_o @ ( image_set_set_int_o @ F @ A3 ) )
          = X ) ) ) ).

% SUP_eqI
thf(fact_813_SUP__mono,axiom,
    ! [A3: set_nat,B3: set_int,F: nat > set_int,G: int > set_int] :
      ( ! [N3: nat] :
          ( ( member_nat @ N3 @ A3 )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ( ord_less_eq_set_int @ ( F @ N3 ) @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ B3 ) ) ) ) ).

% SUP_mono
thf(fact_814_SUP__mono,axiom,
    ! [A3: set_int,B3: set_int,F: int > set_int,G: int > set_int] :
      ( ! [N3: int] :
          ( ( member_int @ N3 @ A3 )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ( ord_less_eq_set_int @ ( F @ N3 ) @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ B3 ) ) ) ) ).

% SUP_mono
thf(fact_815_SUP__mono,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > set_nat,G: nat > set_nat] :
      ( ! [N3: nat] :
          ( ( member_nat @ N3 @ A3 )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ).

% SUP_mono
thf(fact_816_SUP__mono,axiom,
    ! [A3: set_int,B3: set_nat,F: int > set_nat,G: nat > set_nat] :
      ( ! [N3: int] :
          ( ( member_int @ N3 @ A3 )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ).

% SUP_mono
thf(fact_817_SUP__mono,axiom,
    ! [A3: set_set_int,B3: set_int,F: set_int > set_int,G: int > set_int] :
      ( ! [N3: set_int] :
          ( ( member_set_int @ N3 @ A3 )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ( ord_less_eq_set_int @ ( F @ N3 ) @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ B3 ) ) ) ) ).

% SUP_mono
thf(fact_818_SUP__mono,axiom,
    ! [A3: set_nat,B3: set_set_int,F: nat > set_int,G: set_int > set_int] :
      ( ! [N3: nat] :
          ( ( member_nat @ N3 @ A3 )
         => ? [X3: set_int] :
              ( ( member_set_int @ X3 @ B3 )
              & ( ord_less_eq_set_int @ ( F @ N3 ) @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ G @ B3 ) ) ) ) ).

% SUP_mono
thf(fact_819_SUP__mono,axiom,
    ! [A3: set_int,B3: set_set_int,F: int > set_int,G: set_int > set_int] :
      ( ! [N3: int] :
          ( ( member_int @ N3 @ A3 )
         => ? [X3: set_int] :
              ( ( member_set_int @ X3 @ B3 )
              & ( ord_less_eq_set_int @ ( F @ N3 ) @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ G @ B3 ) ) ) ) ).

% SUP_mono
thf(fact_820_SUP__mono,axiom,
    ! [A3: set_set_int,B3: set_nat,F: set_int > set_nat,G: nat > set_nat] :
      ( ! [N3: set_int] :
          ( ( member_set_int @ N3 @ A3 )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B3 )
              & ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ).

% SUP_mono
thf(fact_821_SUP__mono,axiom,
    ! [A3: set_nat,B3: set_int,F: nat > set_set_int,G: int > set_set_int] :
      ( ! [N3: nat] :
          ( ( member_nat @ N3 @ A3 )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ( ord_le4403425263959731960et_int @ ( F @ N3 ) @ ( G @ X3 ) ) ) )
     => ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ ( image_8927401050382224495et_int @ F @ A3 ) ) @ ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ G @ B3 ) ) ) ) ).

% SUP_mono
thf(fact_822_SUP__mono,axiom,
    ! [A3: set_int,B3: set_int,F: int > set_set_int,G: int > set_set_int] :
      ( ! [N3: int] :
          ( ( member_int @ N3 @ A3 )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B3 )
              & ( ord_le4403425263959731960et_int @ ( F @ N3 ) @ ( G @ X3 ) ) ) )
     => ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ F @ A3 ) ) @ ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ G @ B3 ) ) ) ) ).

% SUP_mono
thf(fact_823_SUP__least,axiom,
    ! [A3: set_nat,F: nat > $o,U: $o] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_less_eq_o @ ( F @ I3 ) @ U ) )
     => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_824_SUP__least,axiom,
    ! [A3: set_int,F: int > $o,U: $o] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ( ord_less_eq_o @ ( F @ I3 ) @ U ) )
     => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_int_o @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_825_SUP__least,axiom,
    ! [A3: set_nat,F: nat > set_int,U: set_int] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_less_eq_set_int @ ( F @ I3 ) @ U ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_826_SUP__least,axiom,
    ! [A3: set_int,F: int > set_int,U: set_int] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ( ord_less_eq_set_int @ ( F @ I3 ) @ U ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_827_SUP__least,axiom,
    ! [A3: set_nat,F: nat > set_nat,U: set_nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A3 )
         => ( ord_less_eq_set_nat @ ( F @ I3 ) @ U ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_828_SUP__least,axiom,
    ! [A3: set_int,F: int > set_nat,U: set_nat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A3 )
         => ( ord_less_eq_set_nat @ ( F @ I3 ) @ U ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_829_SUP__least,axiom,
    ! [A3: set_set_int,F: set_int > $o,U: $o] :
      ( ! [I3: set_int] :
          ( ( member_set_int @ I3 @ A3 )
         => ( ord_less_eq_o @ ( F @ I3 ) @ U ) )
     => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_int_o @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_830_SUP__least,axiom,
    ! [A3: set_set_int,F: set_int > set_int,U: set_int] :
      ( ! [I3: set_int] :
          ( ( member_set_int @ I3 @ A3 )
         => ( ord_less_eq_set_int @ ( F @ I3 ) @ U ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_831_SUP__least,axiom,
    ! [A3: set_set_int,F: set_int > set_nat,U: set_nat] :
      ( ! [I3: set_int] :
          ( ( member_set_int @ I3 @ A3 )
         => ( ord_less_eq_set_nat @ ( F @ I3 ) @ U ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_832_SUP__least,axiom,
    ! [A3: set_set_set_int,F: set_set_int > $o,U: $o] :
      ( ! [I3: set_set_int] :
          ( ( member_set_set_int @ I3 @ A3 )
         => ( ord_less_eq_o @ ( F @ I3 ) @ U ) )
     => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_set_int_o @ F @ A3 ) ) @ U ) ) ).

% SUP_least
thf(fact_833_SUP__mono_H,axiom,
    ! [F: set_int > set_int,G: set_int > set_int,A3: set_set_int] :
      ( ! [X5: set_int] : ( ord_less_eq_set_int @ ( F @ X5 ) @ ( G @ X5 ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ G @ A3 ) ) ) ) ).

% SUP_mono'
thf(fact_834_SUP__mono_H,axiom,
    ! [F: int > set_int,G: int > set_int,A3: set_int] :
      ( ! [X5: int] : ( ord_less_eq_set_int @ ( F @ X5 ) @ ( G @ X5 ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ A3 ) ) ) ) ).

% SUP_mono'
thf(fact_835_SUP__mono_H,axiom,
    ! [F: nat > set_nat,G: nat > set_nat,A3: set_nat] :
      ( ! [X5: nat] : ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( G @ X5 ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ A3 ) ) ) ) ).

% SUP_mono'
thf(fact_836_SUP__mono_H,axiom,
    ! [F: int > set_set_int,G: int > set_set_int,A3: set_int] :
      ( ! [X5: int] : ( ord_le4403425263959731960et_int @ ( F @ X5 ) @ ( G @ X5 ) )
     => ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ F @ A3 ) ) @ ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ G @ A3 ) ) ) ) ).

% SUP_mono'
thf(fact_837_SUP__upper,axiom,
    ! [I: nat,A3: set_nat,F: nat > $o] :
      ( ( member_nat @ I @ A3 )
     => ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_838_SUP__upper,axiom,
    ! [I: int,A3: set_int,F: int > $o] :
      ( ( member_int @ I @ A3 )
     => ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_int_o @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_839_SUP__upper,axiom,
    ! [I: nat,A3: set_nat,F: nat > set_int] :
      ( ( member_nat @ I @ A3 )
     => ( ord_less_eq_set_int @ ( F @ I ) @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_840_SUP__upper,axiom,
    ! [I: int,A3: set_int,F: int > set_int] :
      ( ( member_int @ I @ A3 )
     => ( ord_less_eq_set_int @ ( F @ I ) @ ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_841_SUP__upper,axiom,
    ! [I: nat,A3: set_nat,F: nat > set_nat] :
      ( ( member_nat @ I @ A3 )
     => ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_842_SUP__upper,axiom,
    ! [I: int,A3: set_int,F: int > set_nat] :
      ( ( member_int @ I @ A3 )
     => ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_843_SUP__upper,axiom,
    ! [I: set_int,A3: set_set_int,F: set_int > $o] :
      ( ( member_set_int @ I @ A3 )
     => ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_set_int_o @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_844_SUP__upper,axiom,
    ! [I: set_int,A3: set_set_int,F: set_int > set_int] :
      ( ( member_set_int @ I @ A3 )
     => ( ord_less_eq_set_int @ ( F @ I ) @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_845_SUP__upper,axiom,
    ! [I: set_int,A3: set_set_int,F: set_int > set_nat] :
      ( ( member_set_int @ I @ A3 )
     => ( ord_less_eq_set_nat @ ( F @ I ) @ ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_846_SUP__upper,axiom,
    ! [I: set_set_int,A3: set_set_set_int,F: set_set_int > $o] :
      ( ( member_set_set_int @ I @ A3 )
     => ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_set_set_int_o @ F @ A3 ) ) ) ) ).

% SUP_upper
thf(fact_847_SUP__le__iff,axiom,
    ! [F: set_int > set_int,A3: set_set_int,U: set_int] :
      ( ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ A3 ) ) @ U )
      = ( ! [X2: set_int] :
            ( ( member_set_int @ X2 @ A3 )
           => ( ord_less_eq_set_int @ ( F @ X2 ) @ U ) ) ) ) ).

% SUP_le_iff
thf(fact_848_SUP__le__iff,axiom,
    ! [F: int > set_int,A3: set_int,U: set_int] :
      ( ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) ) @ U )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ A3 )
           => ( ord_less_eq_set_int @ ( F @ X2 ) @ U ) ) ) ) ).

% SUP_le_iff
thf(fact_849_SUP__le__iff,axiom,
    ! [F: nat > set_nat,A3: set_nat,U: set_nat] :
      ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ U )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
           => ( ord_less_eq_set_nat @ ( F @ X2 ) @ U ) ) ) ) ).

% SUP_le_iff
thf(fact_850_SUP__le__iff,axiom,
    ! [F: int > set_set_int,A3: set_int,U: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ F @ A3 ) ) @ U )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ A3 )
           => ( ord_le4403425263959731960et_int @ ( F @ X2 ) @ U ) ) ) ) ).

% SUP_le_iff
thf(fact_851_SUP__upper2,axiom,
    ! [I: nat,A3: set_nat,U: $o,F: nat > $o] :
      ( ( member_nat @ I @ A3 )
     => ( ( ord_less_eq_o @ U @ ( F @ I ) )
       => ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_852_SUP__upper2,axiom,
    ! [I: int,A3: set_int,U: $o,F: int > $o] :
      ( ( member_int @ I @ A3 )
     => ( ( ord_less_eq_o @ U @ ( F @ I ) )
       => ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_int_o @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_853_SUP__upper2,axiom,
    ! [I: nat,A3: set_nat,U: set_int,F: nat > set_int] :
      ( ( member_nat @ I @ A3 )
     => ( ( ord_less_eq_set_int @ U @ ( F @ I ) )
       => ( ord_less_eq_set_int @ U @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_854_SUP__upper2,axiom,
    ! [I: int,A3: set_int,U: set_int,F: int > set_int] :
      ( ( member_int @ I @ A3 )
     => ( ( ord_less_eq_set_int @ U @ ( F @ I ) )
       => ( ord_less_eq_set_int @ U @ ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_855_SUP__upper2,axiom,
    ! [I: nat,A3: set_nat,U: set_nat,F: nat > set_nat] :
      ( ( member_nat @ I @ A3 )
     => ( ( ord_less_eq_set_nat @ U @ ( F @ I ) )
       => ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_856_SUP__upper2,axiom,
    ! [I: int,A3: set_int,U: set_nat,F: int > set_nat] :
      ( ( member_int @ I @ A3 )
     => ( ( ord_less_eq_set_nat @ U @ ( F @ I ) )
       => ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_857_SUP__upper2,axiom,
    ! [I: set_int,A3: set_set_int,U: $o,F: set_int > $o] :
      ( ( member_set_int @ I @ A3 )
     => ( ( ord_less_eq_o @ U @ ( F @ I ) )
       => ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_set_int_o @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_858_SUP__upper2,axiom,
    ! [I: set_int,A3: set_set_int,U: set_int,F: set_int > set_int] :
      ( ( member_set_int @ I @ A3 )
     => ( ( ord_less_eq_set_int @ U @ ( F @ I ) )
       => ( ord_less_eq_set_int @ U @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_859_SUP__upper2,axiom,
    ! [I: set_int,A3: set_set_int,U: set_nat,F: set_int > set_nat] :
      ( ( member_set_int @ I @ A3 )
     => ( ( ord_less_eq_set_nat @ U @ ( F @ I ) )
       => ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_860_SUP__upper2,axiom,
    ! [I: set_set_int,A3: set_set_set_int,U: $o,F: set_set_int > $o] :
      ( ( member_set_set_int @ I @ A3 )
     => ( ( ord_less_eq_o @ U @ ( F @ I ) )
       => ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_set_set_int_o @ F @ A3 ) ) ) ) ) ).

% SUP_upper2
thf(fact_861_SUP__lessD,axiom,
    ! [F: nat > $o,A3: set_nat,Y: $o,I: nat] :
      ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A3 ) ) @ Y )
     => ( ( member_nat @ I @ A3 )
       => ( ord_less_o @ ( F @ I ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_862_SUP__lessD,axiom,
    ! [F: int > $o,A3: set_int,Y: $o,I: int] :
      ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_int_o @ F @ A3 ) ) @ Y )
     => ( ( member_int @ I @ A3 )
       => ( ord_less_o @ ( F @ I ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_863_SUP__lessD,axiom,
    ! [F: nat > set_int,A3: set_nat,Y: set_int,I: nat] :
      ( ( ord_less_set_int @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) ) @ Y )
     => ( ( member_nat @ I @ A3 )
       => ( ord_less_set_int @ ( F @ I ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_864_SUP__lessD,axiom,
    ! [F: int > set_int,A3: set_int,Y: set_int,I: int] :
      ( ( ord_less_set_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) ) @ Y )
     => ( ( member_int @ I @ A3 )
       => ( ord_less_set_int @ ( F @ I ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_865_SUP__lessD,axiom,
    ! [F: nat > set_nat,A3: set_nat,Y: set_nat,I: nat] :
      ( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ Y )
     => ( ( member_nat @ I @ A3 )
       => ( ord_less_set_nat @ ( F @ I ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_866_SUP__lessD,axiom,
    ! [F: int > set_nat,A3: set_int,Y: set_nat,I: int] :
      ( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) ) @ Y )
     => ( ( member_int @ I @ A3 )
       => ( ord_less_set_nat @ ( F @ I ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_867_SUP__lessD,axiom,
    ! [F: set_int > $o,A3: set_set_int,Y: $o,I: set_int] :
      ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_set_int_o @ F @ A3 ) ) @ Y )
     => ( ( member_set_int @ I @ A3 )
       => ( ord_less_o @ ( F @ I ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_868_SUP__lessD,axiom,
    ! [F: set_int > set_int,A3: set_set_int,Y: set_int,I: set_int] :
      ( ( ord_less_set_int @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ A3 ) ) @ Y )
     => ( ( member_set_int @ I @ A3 )
       => ( ord_less_set_int @ ( F @ I ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_869_SUP__lessD,axiom,
    ! [F: set_int > set_nat,A3: set_set_int,Y: set_nat,I: set_int] :
      ( ( ord_less_set_nat @ ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ F @ A3 ) ) @ Y )
     => ( ( member_set_int @ I @ A3 )
       => ( ord_less_set_nat @ ( F @ I ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_870_SUP__lessD,axiom,
    ! [F: set_set_int > $o,A3: set_set_set_int,Y: $o,I: set_set_int] :
      ( ( ord_less_o @ ( complete_Sup_Sup_o @ ( image_set_set_int_o @ F @ A3 ) ) @ Y )
     => ( ( member_set_set_int @ I @ A3 )
       => ( ord_less_o @ ( F @ I ) @ Y ) ) ) ).

% SUP_lessD
thf(fact_871_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K4: nat] :
            ( ( P @ K4 )
            & ( ord_less_nat @ K4 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_872_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_873_SUP__eq__iff,axiom,
    ! [I5: set_int,C: $o,F: int > $o] :
      ( ( I5 != bot_bot_set_int )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
       => ( ( ( complete_Sup_Sup_o @ ( image_int_o @ F @ I5 ) )
            = C )
          = ( ! [X2: int] :
                ( ( member_int @ X2 @ I5 )
               => ( ( F @ X2 )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_874_SUP__eq__iff,axiom,
    ! [I5: set_nat,C: $o,F: nat > $o] :
      ( ( I5 != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
       => ( ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I5 ) )
            = C )
          = ( ! [X2: nat] :
                ( ( member_nat @ X2 @ I5 )
               => ( ( F @ X2 )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_875_SUP__eq__iff,axiom,
    ! [I5: set_int,C: set_int,F: int > set_int] :
      ( ( I5 != bot_bot_set_int )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_eq_set_int @ C @ ( F @ I3 ) ) )
       => ( ( ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ I5 ) )
            = C )
          = ( ! [X2: int] :
                ( ( member_int @ X2 @ I5 )
               => ( ( F @ X2 )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_876_SUP__eq__iff,axiom,
    ! [I5: set_nat,C: set_int,F: nat > set_int] :
      ( ( I5 != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ord_less_eq_set_int @ C @ ( F @ I3 ) ) )
       => ( ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ I5 ) )
            = C )
          = ( ! [X2: nat] :
                ( ( member_nat @ X2 @ I5 )
               => ( ( F @ X2 )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_877_SUP__eq__iff,axiom,
    ! [I5: set_int,C: set_nat,F: int > set_nat] :
      ( ( I5 != bot_bot_set_int )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
       => ( ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ I5 ) )
            = C )
          = ( ! [X2: int] :
                ( ( member_int @ X2 @ I5 )
               => ( ( F @ X2 )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_878_SUP__eq__iff,axiom,
    ! [I5: set_nat,C: set_nat,F: nat > set_nat] :
      ( ( I5 != bot_bot_set_nat )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
       => ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I5 ) )
            = C )
          = ( ! [X2: nat] :
                ( ( member_nat @ X2 @ I5 )
               => ( ( F @ X2 )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_879_SUP__eq__iff,axiom,
    ! [I5: set_set_int,C: $o,F: set_int > $o] :
      ( ( I5 != bot_bot_set_set_int )
     => ( ! [I3: set_int] :
            ( ( member_set_int @ I3 @ I5 )
           => ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
       => ( ( ( complete_Sup_Sup_o @ ( image_set_int_o @ F @ I5 ) )
            = C )
          = ( ! [X2: set_int] :
                ( ( member_set_int @ X2 @ I5 )
               => ( ( F @ X2 )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_880_SUP__eq__iff,axiom,
    ! [I5: set_set_int,C: set_int,F: set_int > set_int] :
      ( ( I5 != bot_bot_set_set_int )
     => ( ! [I3: set_int] :
            ( ( member_set_int @ I3 @ I5 )
           => ( ord_less_eq_set_int @ C @ ( F @ I3 ) ) )
       => ( ( ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ I5 ) )
            = C )
          = ( ! [X2: set_int] :
                ( ( member_set_int @ X2 @ I5 )
               => ( ( F @ X2 )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_881_SUP__eq__iff,axiom,
    ! [I5: set_set_int,C: set_nat,F: set_int > set_nat] :
      ( ( I5 != bot_bot_set_set_int )
     => ( ! [I3: set_int] :
            ( ( member_set_int @ I3 @ I5 )
           => ( ord_less_eq_set_nat @ C @ ( F @ I3 ) ) )
       => ( ( ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ F @ I5 ) )
            = C )
          = ( ! [X2: set_int] :
                ( ( member_set_int @ X2 @ I5 )
               => ( ( F @ X2 )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_882_SUP__eq__iff,axiom,
    ! [I5: set_set_set_int,C: $o,F: set_set_int > $o] :
      ( ( I5 != bot_bo2384636101374064866et_int )
     => ( ! [I3: set_set_int] :
            ( ( member_set_set_int @ I3 @ I5 )
           => ( ord_less_eq_o @ C @ ( F @ I3 ) ) )
       => ( ( ( complete_Sup_Sup_o @ ( image_set_set_int_o @ F @ I5 ) )
            = C )
          = ( ! [X2: set_set_int] :
                ( ( member_set_set_int @ X2 @ I5 )
               => ( ( F @ X2 )
                  = C ) ) ) ) ) ) ).

% SUP_eq_iff
thf(fact_883_SUP__subset__mono,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > $o,G: nat > $o] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A3 )
           => ( ord_less_eq_o @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A3 ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_884_SUP__subset__mono,axiom,
    ! [A3: set_int,B3: set_int,F: int > $o,G: int > $o] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A3 )
           => ( ord_less_eq_o @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_int_o @ F @ A3 ) ) @ ( complete_Sup_Sup_o @ ( image_int_o @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_885_SUP__subset__mono,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > set_int,G: nat > set_int] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A3 )
           => ( ord_less_eq_set_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_886_SUP__subset__mono,axiom,
    ! [A3: set_int,B3: set_int,F: int > set_int,G: int > set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A3 )
           => ( ord_less_eq_set_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_int_set_int @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_887_SUP__subset__mono,axiom,
    ! [A3: set_nat,B3: set_nat,F: nat > set_nat,G: nat > set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A3 )
           => ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_888_SUP__subset__mono,axiom,
    ! [A3: set_int,B3: set_int,F: int > set_nat,G: int > set_nat] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A3 )
           => ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_889_SUP__subset__mono,axiom,
    ! [A3: set_set_int,B3: set_set_int,F: set_int > $o,G: set_int > $o] :
      ( ( ord_le4403425263959731960et_int @ A3 @ B3 )
     => ( ! [X5: set_int] :
            ( ( member_set_int @ X5 @ A3 )
           => ( ord_less_eq_o @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_int_o @ F @ A3 ) ) @ ( complete_Sup_Sup_o @ ( image_set_int_o @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_890_SUP__subset__mono,axiom,
    ! [A3: set_set_int,B3: set_set_int,F: set_int > set_int,G: set_int > set_int] :
      ( ( ord_le4403425263959731960et_int @ A3 @ B3 )
     => ( ! [X5: set_int] :
            ( ( member_set_int @ X5 @ A3 )
           => ( ord_less_eq_set_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ A3 ) ) @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_891_SUP__subset__mono,axiom,
    ! [A3: set_set_int,B3: set_set_int,F: set_int > set_nat,G: set_int > set_nat] :
      ( ( ord_le4403425263959731960et_int @ A3 @ B3 )
     => ( ! [X5: set_int] :
            ( ( member_set_int @ X5 @ A3 )
           => ( ord_less_eq_set_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ F @ A3 ) ) @ ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_892_SUP__subset__mono,axiom,
    ! [A3: set_set_set_int,B3: set_set_set_int,F: set_set_int > $o,G: set_set_int > $o] :
      ( ( ord_le4317611570275147438et_int @ A3 @ B3 )
     => ( ! [X5: set_set_int] :
            ( ( member_set_set_int @ X5 @ A3 )
           => ( ord_less_eq_o @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_set_set_int_o @ F @ A3 ) ) @ ( complete_Sup_Sup_o @ ( image_set_set_int_o @ G @ B3 ) ) ) ) ) ).

% SUP_subset_mono
thf(fact_893_SUP__constant,axiom,
    ! [C: $o,A3: set_int] :
      ( ( complete_Sup_Sup_o
        @ ( image_int_o
          @ ^ [Y2: int] : C
          @ A3 ) )
      = ( ( ( A3 = bot_bot_set_int )
         => bot_bot_o )
        & ( ( A3 != bot_bot_set_int )
         => C ) ) ) ).

% SUP_constant
thf(fact_894_SUP__constant,axiom,
    ! [C: $o,A3: set_nat] :
      ( ( complete_Sup_Sup_o
        @ ( image_nat_o
          @ ^ [Y2: nat] : C
          @ A3 ) )
      = ( ( ( A3 = bot_bot_set_nat )
         => bot_bot_o )
        & ( ( A3 != bot_bot_set_nat )
         => C ) ) ) ).

% SUP_constant
thf(fact_895_SUP__constant,axiom,
    ! [A3: set_int,C: set_int] :
      ( ( ( A3 = bot_bot_set_int )
       => ( ( comple3221217463730067765et_int
            @ ( image_int_set_int
              @ ^ [Y2: int] : C
              @ A3 ) )
          = bot_bot_set_int ) )
      & ( ( A3 != bot_bot_set_int )
       => ( ( comple3221217463730067765et_int
            @ ( image_int_set_int
              @ ^ [Y2: int] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_896_SUP__constant,axiom,
    ! [A3: set_nat,C: set_int] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple3221217463730067765et_int
            @ ( image_nat_set_int
              @ ^ [Y2: nat] : C
              @ A3 ) )
          = bot_bot_set_int ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple3221217463730067765et_int
            @ ( image_nat_set_int
              @ ^ [Y2: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_897_SUP__constant,axiom,
    ! [A3: set_int,C: set_nat] :
      ( ( ( A3 = bot_bot_set_int )
       => ( ( comple7399068483239264473et_nat
            @ ( image_int_set_nat
              @ ^ [Y2: int] : C
              @ A3 ) )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_int )
       => ( ( comple7399068483239264473et_nat
            @ ( image_int_set_nat
              @ ^ [Y2: int] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_898_SUP__constant,axiom,
    ! [A3: set_nat,C: set_nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [Y2: nat] : C
              @ A3 ) )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [Y2: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_899_SUP__constant,axiom,
    ! [C: $o,A3: set_set_int] :
      ( ( complete_Sup_Sup_o
        @ ( image_set_int_o
          @ ^ [Y2: set_int] : C
          @ A3 ) )
      = ( ( ( A3 = bot_bot_set_set_int )
         => bot_bot_o )
        & ( ( A3 != bot_bot_set_set_int )
         => C ) ) ) ).

% SUP_constant
thf(fact_900_SUP__constant,axiom,
    ! [A3: set_set_int,C: set_int] :
      ( ( ( A3 = bot_bot_set_set_int )
       => ( ( comple3221217463730067765et_int
            @ ( image_524474410958335435et_int
              @ ^ [Y2: set_int] : C
              @ A3 ) )
          = bot_bot_set_int ) )
      & ( ( A3 != bot_bot_set_set_int )
       => ( ( comple3221217463730067765et_int
            @ ( image_524474410958335435et_int
              @ ^ [Y2: set_int] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_901_SUP__constant,axiom,
    ! [A3: set_set_int,C: set_nat] :
      ( ( ( A3 = bot_bot_set_set_int )
       => ( ( comple7399068483239264473et_nat
            @ ( image_4702325430467532143et_nat
              @ ^ [Y2: set_int] : C
              @ A3 ) )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_set_int )
       => ( ( comple7399068483239264473et_nat
            @ ( image_4702325430467532143et_nat
              @ ^ [Y2: set_int] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_902_SUP__constant,axiom,
    ! [A3: set_int,C: set_set_int] :
      ( ( ( A3 = bot_bot_set_int )
       => ( ( comple7281953568134767595et_int
            @ ( image_6617583118289273547et_int
              @ ^ [Y2: int] : C
              @ A3 ) )
          = bot_bot_set_set_int ) )
      & ( ( A3 != bot_bot_set_int )
       => ( ( comple7281953568134767595et_int
            @ ( image_6617583118289273547et_int
              @ ^ [Y2: int] : C
              @ A3 ) )
          = C ) ) ) ).

% SUP_constant
thf(fact_903_SUP__empty,axiom,
    ! [F: int > $o] :
      ( ( complete_Sup_Sup_o @ ( image_int_o @ F @ bot_bot_set_int ) )
      = bot_bot_o ) ).

% SUP_empty
thf(fact_904_SUP__empty,axiom,
    ! [F: nat > $o] :
      ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ bot_bot_set_nat ) )
      = bot_bot_o ) ).

% SUP_empty
thf(fact_905_SUP__empty,axiom,
    ! [F: int > set_int] :
      ( ( comple3221217463730067765et_int @ ( image_int_set_int @ F @ bot_bot_set_int ) )
      = bot_bot_set_int ) ).

% SUP_empty
thf(fact_906_SUP__empty,axiom,
    ! [F: nat > set_int] :
      ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ F @ bot_bot_set_nat ) )
      = bot_bot_set_int ) ).

% SUP_empty
thf(fact_907_SUP__empty,axiom,
    ! [F: int > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ F @ bot_bot_set_int ) )
      = bot_bot_set_nat ) ).

% SUP_empty
thf(fact_908_SUP__empty,axiom,
    ! [F: nat > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) )
      = bot_bot_set_nat ) ).

% SUP_empty
thf(fact_909_SUP__empty,axiom,
    ! [F: set_int > $o] :
      ( ( complete_Sup_Sup_o @ ( image_set_int_o @ F @ bot_bot_set_set_int ) )
      = bot_bot_o ) ).

% SUP_empty
thf(fact_910_SUP__empty,axiom,
    ! [F: set_int > set_int] :
      ( ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ F @ bot_bot_set_set_int ) )
      = bot_bot_set_int ) ).

% SUP_empty
thf(fact_911_SUP__empty,axiom,
    ! [F: set_int > set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ F @ bot_bot_set_set_int ) )
      = bot_bot_set_nat ) ).

% SUP_empty
thf(fact_912_SUP__empty,axiom,
    ! [F: int > set_set_int] :
      ( ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ F @ bot_bot_set_int ) )
      = bot_bot_set_set_int ) ).

% SUP_empty
thf(fact_913_int_Oadd_Osurj__const__mult,axiom,
    ! [A: int] :
      ( ( member_int @ A @ top_top_set_int )
     => ( ( image_int_int @ ( plus_plus_int @ A ) @ top_top_set_int )
        = top_top_set_int ) ) ).

% int.add.surj_const_mult
thf(fact_914_lessThan__empty__iff,axiom,
    ! [N2: nat] :
      ( ( ( set_ord_lessThan_nat @ N2 )
        = bot_bot_set_nat )
      = ( N2 = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_915_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_916_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_917_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_918_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_919_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_920_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_921_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_922_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_923_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_924_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_925_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_926_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] :
            ( ! [Z3: int > set_int] :
                ( ( member_int_set_int @ Z3 @ A3 )
               => ( ord_le4076609313706656525et_int @ Z3 @ Y4 ) )
           => ( ord_le4076609313706656525et_int @ X @ Y4 ) )
       => ( ( comple465308567373470592et_int @ A3 )
          = X ) ) ) ).

% Sup_eqI
thf(fact_927_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] :
            ( ! [Z3: set_int] :
                ( ( member_set_int @ Z3 @ A3 )
               => ( ord_less_eq_set_int @ Z3 @ Y4 ) )
           => ( ord_less_eq_set_int @ X @ Y4 ) )
       => ( ( comple3221217463730067765et_int @ A3 )
          = X ) ) ) ).

% Sup_eqI
thf(fact_928_Sup__eqI,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( ! [Y4: set_nat] :
          ( ( member_set_nat @ Y4 @ A3 )
         => ( ord_less_eq_set_nat @ Y4 @ X ) )
     => ( ! [Y4: set_nat] :
            ( ! [Z3: set_nat] :
                ( ( member_set_nat @ Z3 @ A3 )
               => ( ord_less_eq_set_nat @ Z3 @ Y4 ) )
           => ( ord_less_eq_set_nat @ X @ Y4 ) )
       => ( ( comple7399068483239264473et_nat @ A3 )
          = X ) ) ) ).

% Sup_eqI
thf(fact_929_Sup__eqI,axiom,
    ! [A3: set_o,X: $o] :
      ( ! [Y4: $o] :
          ( ( member_o @ Y4 @ A3 )
         => ( ord_less_eq_o @ Y4 @ X ) )
     => ( ! [Y4: $o] :
            ( ! [Z3: $o] :
                ( ( member_o @ Z3 @ A3 )
               => ( ord_less_eq_o @ Z3 @ Y4 ) )
           => ( ord_less_eq_o @ X @ Y4 ) )
       => ( ( complete_Sup_Sup_o @ A3 )
          = X ) ) ) ).

% Sup_eqI
thf(fact_930_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] :
            ( ! [Z3: set_set_int] :
                ( ( member_set_set_int @ Z3 @ A3 )
               => ( ord_le4403425263959731960et_int @ Z3 @ Y4 ) )
           => ( ord_le4403425263959731960et_int @ X @ Y4 ) )
       => ( ( comple7281953568134767595et_int @ A3 )
          = X ) ) ) ).

% Sup_eqI
thf(fact_931_Sup__mono,axiom,
    ! [A3: set_int_set_int,B3: set_int_set_int] :
      ( ! [A5: int > set_int] :
          ( ( member_int_set_int @ A5 @ A3 )
         => ? [X3: int > set_int] :
              ( ( member_int_set_int @ X3 @ B3 )
              & ( ord_le4076609313706656525et_int @ A5 @ X3 ) ) )
     => ( ord_le4076609313706656525et_int @ ( comple465308567373470592et_int @ A3 ) @ ( comple465308567373470592et_int @ B3 ) ) ) ).

% Sup_mono
thf(fact_932_Sup__mono,axiom,
    ! [A3: set_set_int,B3: set_set_int] :
      ( ! [A5: set_int] :
          ( ( member_set_int @ A5 @ A3 )
         => ? [X3: set_int] :
              ( ( member_set_int @ X3 @ B3 )
              & ( ord_less_eq_set_int @ A5 @ X3 ) ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ ( comple3221217463730067765et_int @ B3 ) ) ) ).

% Sup_mono
thf(fact_933_Sup__mono,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ! [A5: set_nat] :
          ( ( member_set_nat @ A5 @ A3 )
         => ? [X3: set_nat] :
              ( ( member_set_nat @ X3 @ B3 )
              & ( ord_less_eq_set_nat @ A5 @ X3 ) ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ ( comple7399068483239264473et_nat @ B3 ) ) ) ).

% Sup_mono
thf(fact_934_Sup__mono,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ! [A5: $o] :
          ( ( member_o @ A5 @ A3 )
         => ? [X3: $o] :
              ( ( member_o @ X3 @ B3 )
              & ( ord_less_eq_o @ A5 @ X3 ) ) )
     => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A3 ) @ ( complete_Sup_Sup_o @ B3 ) ) ) ).

% Sup_mono
thf(fact_935_Sup__mono,axiom,
    ! [A3: set_set_set_int,B3: set_set_set_int] :
      ( ! [A5: set_set_int] :
          ( ( member_set_set_int @ A5 @ A3 )
         => ? [X3: set_set_int] :
              ( ( member_set_set_int @ X3 @ B3 )
              & ( ord_le4403425263959731960et_int @ A5 @ X3 ) ) )
     => ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A3 ) @ ( comple7281953568134767595et_int @ B3 ) ) ) ).

% Sup_mono
thf(fact_936_Sup__least,axiom,
    ! [A3: set_int_set_int,Z: int > set_int] :
      ( ! [X5: int > set_int] :
          ( ( member_int_set_int @ X5 @ A3 )
         => ( ord_le4076609313706656525et_int @ X5 @ Z ) )
     => ( ord_le4076609313706656525et_int @ ( comple465308567373470592et_int @ A3 ) @ Z ) ) ).

% Sup_least
thf(fact_937_Sup__least,axiom,
    ! [A3: set_set_int,Z: set_int] :
      ( ! [X5: set_int] :
          ( ( member_set_int @ X5 @ A3 )
         => ( ord_less_eq_set_int @ X5 @ Z ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ Z ) ) ).

% Sup_least
thf(fact_938_Sup__least,axiom,
    ! [A3: set_set_nat,Z: set_nat] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ A3 )
         => ( ord_less_eq_set_nat @ X5 @ Z ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ Z ) ) ).

% Sup_least
thf(fact_939_Sup__least,axiom,
    ! [A3: set_o,Z: $o] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A3 )
         => ( ord_less_eq_o @ X5 @ Z ) )
     => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A3 ) @ Z ) ) ).

% Sup_least
thf(fact_940_Sup__least,axiom,
    ! [A3: set_set_set_int,Z: set_set_int] :
      ( ! [X5: set_set_int] :
          ( ( member_set_set_int @ X5 @ A3 )
         => ( ord_le4403425263959731960et_int @ X5 @ Z ) )
     => ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A3 ) @ Z ) ) ).

% Sup_least
thf(fact_941_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_942_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_943_Sup__upper,axiom,
    ! [X: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ X @ A3 )
     => ( ord_less_eq_set_nat @ X @ ( comple7399068483239264473et_nat @ A3 ) ) ) ).

% Sup_upper
thf(fact_944_Sup__upper,axiom,
    ! [X: $o,A3: set_o] :
      ( ( member_o @ X @ A3 )
     => ( ord_less_eq_o @ X @ ( complete_Sup_Sup_o @ A3 ) ) ) ).

% Sup_upper
thf(fact_945_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_946_Sup__le__iff,axiom,
    ! [A3: set_set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ B )
      = ( ! [X2: set_int] :
            ( ( member_set_int @ X2 @ A3 )
           => ( ord_less_eq_set_int @ X2 @ B ) ) ) ) ).

% Sup_le_iff
thf(fact_947_Sup__le__iff,axiom,
    ! [A3: set_set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ B )
      = ( ! [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A3 )
           => ( ord_less_eq_set_nat @ X2 @ B ) ) ) ) ).

% Sup_le_iff
thf(fact_948_Sup__le__iff,axiom,
    ! [A3: set_o,B: $o] :
      ( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A3 ) @ B )
      = ( ! [X2: $o] :
            ( ( member_o @ X2 @ A3 )
           => ( ord_less_eq_o @ X2 @ B ) ) ) ) ).

% Sup_le_iff
thf(fact_949_Sup__le__iff,axiom,
    ! [A3: set_set_set_int,B: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A3 ) @ B )
      = ( ! [X2: set_set_int] :
            ( ( member_set_set_int @ X2 @ A3 )
           => ( ord_le4403425263959731960et_int @ X2 @ B ) ) ) ) ).

% Sup_le_iff
thf(fact_950_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_951_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_952_Sup__upper2,axiom,
    ! [U: set_nat,A3: set_set_nat,V: set_nat] :
      ( ( member_set_nat @ U @ A3 )
     => ( ( ord_less_eq_set_nat @ V @ U )
       => ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ).

% Sup_upper2
thf(fact_953_Sup__upper2,axiom,
    ! [U: $o,A3: set_o,V: $o] :
      ( ( member_o @ U @ A3 )
     => ( ( ord_less_eq_o @ V @ U )
       => ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A3 ) ) ) ) ).

% Sup_upper2
thf(fact_954_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_955_Union__UNIV,axiom,
    ( ( comple3221217463730067765et_int @ top_top_set_set_int )
    = top_top_set_int ) ).

% Union_UNIV
thf(fact_956_Union__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ top_top_set_set_nat )
    = top_top_set_nat ) ).

% Union_UNIV
thf(fact_957_Union__UNIV,axiom,
    ( ( comple7281953568134767595et_int @ top_to5524576366173240574et_int )
    = top_top_set_set_int ) ).

% Union_UNIV
thf(fact_958_Union__mono,axiom,
    ! [A3: set_set_int,B3: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ A3 @ B3 )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ ( comple3221217463730067765et_int @ B3 ) ) ) ).

% Union_mono
thf(fact_959_Union__mono,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ ( comple7399068483239264473et_nat @ B3 ) ) ) ).

% Union_mono
thf(fact_960_Union__mono,axiom,
    ! [A3: set_set_set_int,B3: set_set_set_int] :
      ( ( ord_le4317611570275147438et_int @ A3 @ B3 )
     => ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A3 ) @ ( comple7281953568134767595et_int @ B3 ) ) ) ).

% Union_mono
thf(fact_961_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_962_Union__least,axiom,
    ! [A3: set_set_nat,C2: set_nat] :
      ( ! [X8: set_nat] :
          ( ( member_set_nat @ X8 @ A3 )
         => ( ord_less_eq_set_nat @ X8 @ C2 ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ C2 ) ) ).

% Union_least
thf(fact_963_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_964_Union__upper,axiom,
    ! [B3: set_int,A3: set_set_int] :
      ( ( member_set_int @ B3 @ A3 )
     => ( ord_less_eq_set_int @ B3 @ ( comple3221217463730067765et_int @ A3 ) ) ) ).

% Union_upper
thf(fact_965_Union__upper,axiom,
    ! [B3: set_nat,A3: set_set_nat] :
      ( ( member_set_nat @ B3 @ A3 )
     => ( ord_less_eq_set_nat @ B3 @ ( comple7399068483239264473et_nat @ A3 ) ) ) ).

% Union_upper
thf(fact_966_Union__upper,axiom,
    ! [B3: set_set_int,A3: set_set_set_int] :
      ( ( member_set_set_int @ B3 @ A3 )
     => ( ord_le4403425263959731960et_int @ B3 @ ( comple7281953568134767595et_int @ A3 ) ) ) ).

% Union_upper
thf(fact_967_Union__subsetI,axiom,
    ! [A3: set_set_int,B3: set_set_int] :
      ( ! [X5: set_int] :
          ( ( member_set_int @ X5 @ A3 )
         => ? [Y5: set_int] :
              ( ( member_set_int @ Y5 @ B3 )
              & ( ord_less_eq_set_int @ X5 @ Y5 ) ) )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ ( comple3221217463730067765et_int @ B3 ) ) ) ).

% Union_subsetI
thf(fact_968_Union__subsetI,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ A3 )
         => ? [Y5: set_nat] :
              ( ( member_set_nat @ Y5 @ B3 )
              & ( ord_less_eq_set_nat @ X5 @ Y5 ) ) )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ ( comple7399068483239264473et_nat @ B3 ) ) ) ).

% Union_subsetI
thf(fact_969_Union__subsetI,axiom,
    ! [A3: set_set_set_int,B3: set_set_set_int] :
      ( ! [X5: set_set_int] :
          ( ( member_set_set_int @ X5 @ A3 )
         => ? [Y5: set_set_int] :
              ( ( member_set_set_int @ Y5 @ B3 )
              & ( ord_le4403425263959731960et_int @ X5 @ Y5 ) ) )
     => ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A3 ) @ ( comple7281953568134767595et_int @ B3 ) ) ) ).

% Union_subsetI
thf(fact_970_Union__empty,axiom,
    ( ( comple3221217463730067765et_int @ bot_bot_set_set_int )
    = bot_bot_set_int ) ).

% Union_empty
thf(fact_971_Union__empty,axiom,
    ( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
    = bot_bot_set_nat ) ).

% Union_empty
thf(fact_972_Union__empty,axiom,
    ( ( comple7281953568134767595et_int @ bot_bo2384636101374064866et_int )
    = bot_bot_set_set_int ) ).

% Union_empty
thf(fact_973_Union__empty__conv,axiom,
    ! [A3: set_set_int] :
      ( ( ( comple3221217463730067765et_int @ A3 )
        = bot_bot_set_int )
      = ( ! [X2: set_int] :
            ( ( member_set_int @ X2 @ A3 )
           => ( X2 = bot_bot_set_int ) ) ) ) ).

% Union_empty_conv
thf(fact_974_Union__empty__conv,axiom,
    ! [A3: set_set_nat] :
      ( ( ( comple7399068483239264473et_nat @ A3 )
        = bot_bot_set_nat )
      = ( ! [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A3 )
           => ( X2 = bot_bot_set_nat ) ) ) ) ).

% Union_empty_conv
thf(fact_975_Union__empty__conv,axiom,
    ! [A3: set_set_set_int] :
      ( ( ( comple7281953568134767595et_int @ A3 )
        = bot_bot_set_set_int )
      = ( ! [X2: set_set_int] :
            ( ( member_set_set_int @ X2 @ A3 )
           => ( X2 = bot_bot_set_set_int ) ) ) ) ).

% Union_empty_conv
thf(fact_976_empty__Union__conv,axiom,
    ! [A3: set_set_int] :
      ( ( bot_bot_set_int
        = ( comple3221217463730067765et_int @ A3 ) )
      = ( ! [X2: set_int] :
            ( ( member_set_int @ X2 @ A3 )
           => ( X2 = bot_bot_set_int ) ) ) ) ).

% empty_Union_conv
thf(fact_977_empty__Union__conv,axiom,
    ! [A3: set_set_nat] :
      ( ( bot_bot_set_nat
        = ( comple7399068483239264473et_nat @ A3 ) )
      = ( ! [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A3 )
           => ( X2 = bot_bot_set_nat ) ) ) ) ).

% empty_Union_conv
thf(fact_978_empty__Union__conv,axiom,
    ! [A3: set_set_set_int] :
      ( ( bot_bot_set_set_int
        = ( comple7281953568134767595et_int @ A3 ) )
      = ( ! [X2: set_set_int] :
            ( ( member_set_set_int @ X2 @ A3 )
           => ( X2 = bot_bot_set_set_int ) ) ) ) ).

% empty_Union_conv
thf(fact_979_Sup__subset__mono,axiom,
    ! [A3: set_set_int,B3: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ A3 @ B3 )
     => ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A3 ) @ ( comple3221217463730067765et_int @ B3 ) ) ) ).

% Sup_subset_mono
thf(fact_980_Sup__subset__mono,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
     => ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A3 ) @ ( comple7399068483239264473et_nat @ B3 ) ) ) ).

% Sup_subset_mono
thf(fact_981_Sup__subset__mono,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ B3 )
     => ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A3 ) @ ( complete_Sup_Sup_o @ B3 ) ) ) ).

% Sup_subset_mono
thf(fact_982_Sup__subset__mono,axiom,
    ! [A3: set_set_set_int,B3: set_set_set_int] :
      ( ( ord_le4317611570275147438et_int @ A3 @ B3 )
     => ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A3 ) @ ( comple7281953568134767595et_int @ B3 ) ) ) ).

% Sup_subset_mono
thf(fact_983_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_984_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_985_less__eq__Sup,axiom,
    ! [A3: set_set_nat,U: set_nat] :
      ( ! [V2: set_nat] :
          ( ( member_set_nat @ V2 @ A3 )
         => ( ord_less_eq_set_nat @ U @ V2 ) )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ord_less_eq_set_nat @ U @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ).

% less_eq_Sup
thf(fact_986_less__eq__Sup,axiom,
    ! [A3: set_o,U: $o] :
      ( ! [V2: $o] :
          ( ( member_o @ V2 @ A3 )
         => ( ord_less_eq_o @ U @ V2 ) )
     => ( ( A3 != bot_bot_set_o )
       => ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ A3 ) ) ) ) ).

% less_eq_Sup
thf(fact_987_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_988_cSUP__const,axiom,
    ! [A3: set_int,C: int] :
      ( ( A3 != bot_bot_set_int )
     => ( ( complete_Sup_Sup_int
          @ ( image_int_int
            @ ^ [X2: int] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_989_cSUP__const,axiom,
    ! [A3: set_int,C: nat] :
      ( ( A3 != bot_bot_set_int )
     => ( ( complete_Sup_Sup_nat
          @ ( image_int_nat
            @ ^ [X2: int] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_990_cSUP__const,axiom,
    ! [A3: set_nat,C: nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( complete_Sup_Sup_nat
          @ ( image_nat_nat
            @ ^ [X2: nat] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_991_cSUP__const,axiom,
    ! [A3: set_int,C: $o] :
      ( ( A3 != bot_bot_set_int )
     => ( ( complete_Sup_Sup_o
          @ ( image_int_o
            @ ^ [X2: int] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_992_cSUP__const,axiom,
    ! [A3: set_nat,C: $o] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( complete_Sup_Sup_o
          @ ( image_nat_o
            @ ^ [X2: nat] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_993_cSUP__const,axiom,
    ! [A3: set_int,C: set_int] :
      ( ( A3 != bot_bot_set_int )
     => ( ( comple3221217463730067765et_int
          @ ( image_int_set_int
            @ ^ [X2: int] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_994_cSUP__const,axiom,
    ! [A3: set_nat,C: set_int] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( comple3221217463730067765et_int
          @ ( image_nat_set_int
            @ ^ [X2: nat] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_995_cSUP__const,axiom,
    ! [A3: set_set_int,C: nat] :
      ( ( A3 != bot_bot_set_set_int )
     => ( ( complete_Sup_Sup_nat
          @ ( image_set_int_nat
            @ ^ [X2: set_int] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_996_cSUP__const,axiom,
    ! [A3: set_int,C: set_nat] :
      ( ( A3 != bot_bot_set_int )
     => ( ( comple7399068483239264473et_nat
          @ ( image_int_set_nat
            @ ^ [X2: int] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_997_cSUP__const,axiom,
    ! [A3: set_nat,C: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ( comple7399068483239264473et_nat
          @ ( image_nat_set_nat
            @ ^ [X2: nat] : C
            @ A3 ) )
        = C ) ) ).

% cSUP_const
thf(fact_998_surj__plus__right,axiom,
    ! [A: int] :
      ( ( image_int_int
        @ ^ [B4: int] : ( plus_plus_int @ B4 @ A )
        @ top_top_set_int )
      = top_top_set_int ) ).

% surj_plus_right
thf(fact_999_surj__uminus,axiom,
    ( ( image_int_int @ uminus_uminus_int @ top_top_set_int )
    = top_top_set_int ) ).

% surj_uminus
thf(fact_1000_i_Oadd_Oinj__on__g,axiom,
    ! [H2: set_int,A: 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 ) ) ) ) )
     => ( ( 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 ) ) ) ) )
       => ( inj_on_int_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 ) ) ) @ Y2 @ A )
          @ H2 ) ) ) ).

% i.add.inj_on_g
thf(fact_1001_inj__on__empty,axiom,
    ! [F: int > int] : ( inj_on_int_int @ F @ bot_bot_set_int ) ).

% inj_on_empty
thf(fact_1002_inj__on__empty,axiom,
    ! [F: int > set_int] : ( inj_on_int_set_int @ F @ bot_bot_set_int ) ).

% inj_on_empty
thf(fact_1003_inj__on__empty,axiom,
    ! [F: set_int > set_int] : ( inj_on6435365835345961783et_int @ F @ bot_bot_set_set_int ) ).

% inj_on_empty
thf(fact_1004_inj__uminus,axiom,
    ! [A3: set_int] : ( inj_on_int_int @ uminus_uminus_int @ A3 ) ).

% inj_uminus
thf(fact_1005_UN__ball__bex__simps_I4_J,axiom,
    ! [B3: set_int > set_int,A3: set_set_int,P: int > $o] :
      ( ( ? [X2: int] :
            ( ( member_int @ X2 @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ B3 @ A3 ) ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: set_int] :
            ( ( member_set_int @ X2 @ A3 )
            & ? [Y2: int] :
                ( ( member_int @ Y2 @ ( B3 @ X2 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(4)
thf(fact_1006_UN__ball__bex__simps_I4_J,axiom,
    ! [B3: int > set_int,A3: set_int,P: int > $o] :
      ( ( ? [X2: int] :
            ( ( member_int @ X2 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A3 ) ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: int] :
            ( ( member_int @ X2 @ A3 )
            & ? [Y2: int] :
                ( ( member_int @ Y2 @ ( B3 @ X2 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(4)
thf(fact_1007_UN__ball__bex__simps_I4_J,axiom,
    ! [B3: nat > set_nat,A3: set_nat,P: nat > $o] :
      ( ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
            & ? [Y2: nat] :
                ( ( member_nat @ Y2 @ ( B3 @ X2 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(4)
thf(fact_1008_UN__ball__bex__simps_I4_J,axiom,
    ! [B3: int > set_set_int,A3: set_int,P: set_int > $o] :
      ( ( ? [X2: set_int] :
            ( ( member_set_int @ X2 @ ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ B3 @ A3 ) ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: int] :
            ( ( member_int @ X2 @ A3 )
            & ? [Y2: set_int] :
                ( ( member_set_int @ Y2 @ ( B3 @ X2 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(4)
thf(fact_1009_UN__ball__bex__simps_I2_J,axiom,
    ! [B3: set_int > set_int,A3: set_set_int,P: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int @ X2 @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ B3 @ A3 ) ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: set_int] :
            ( ( member_set_int @ X2 @ A3 )
           => ! [Y2: int] :
                ( ( member_int @ Y2 @ ( B3 @ X2 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(2)
thf(fact_1010_UN__ball__bex__simps_I2_J,axiom,
    ! [B3: int > set_int,A3: set_int,P: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int @ X2 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A3 ) ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ A3 )
           => ! [Y2: int] :
                ( ( member_int @ Y2 @ ( B3 @ X2 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(2)
thf(fact_1011_UN__ball__bex__simps_I2_J,axiom,
    ! [B3: nat > set_nat,A3: set_nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
           => ! [Y2: nat] :
                ( ( member_nat @ Y2 @ ( B3 @ X2 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(2)
thf(fact_1012_UN__ball__bex__simps_I2_J,axiom,
    ! [B3: int > set_set_int,A3: set_int,P: set_int > $o] :
      ( ( ! [X2: set_int] :
            ( ( member_set_int @ X2 @ ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ B3 @ A3 ) ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ A3 )
           => ! [Y2: set_int] :
                ( ( member_set_int @ Y2 @ ( B3 @ X2 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% UN_ball_bex_simps(2)
thf(fact_1013_bex__UN,axiom,
    ! [B3: set_int > set_int,A3: set_set_int,P: int > $o] :
      ( ( ? [X2: int] :
            ( ( member_int @ X2 @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ B3 @ A3 ) ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: set_int] :
            ( ( member_set_int @ X2 @ A3 )
            & ? [Y2: int] :
                ( ( member_int @ Y2 @ ( B3 @ X2 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% bex_UN
thf(fact_1014_bex__UN,axiom,
    ! [B3: int > set_int,A3: set_int,P: int > $o] :
      ( ( ? [X2: int] :
            ( ( member_int @ X2 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A3 ) ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: int] :
            ( ( member_int @ X2 @ A3 )
            & ? [Y2: int] :
                ( ( member_int @ Y2 @ ( B3 @ X2 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% bex_UN
thf(fact_1015_bex__UN,axiom,
    ! [B3: nat > set_nat,A3: set_nat,P: nat > $o] :
      ( ( ? [X2: nat] :
            ( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
            & ? [Y2: nat] :
                ( ( member_nat @ Y2 @ ( B3 @ X2 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% bex_UN
thf(fact_1016_bex__UN,axiom,
    ! [B3: int > set_set_int,A3: set_int,P: set_int > $o] :
      ( ( ? [X2: set_int] :
            ( ( member_set_int @ X2 @ ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ B3 @ A3 ) ) )
            & ( P @ X2 ) ) )
      = ( ? [X2: int] :
            ( ( member_int @ X2 @ A3 )
            & ? [Y2: set_int] :
                ( ( member_set_int @ Y2 @ ( B3 @ X2 ) )
                & ( P @ Y2 ) ) ) ) ) ).

% bex_UN
thf(fact_1017_ball__UN,axiom,
    ! [B3: set_int > set_int,A3: set_set_int,P: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int @ X2 @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ B3 @ A3 ) ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: set_int] :
            ( ( member_set_int @ X2 @ A3 )
           => ! [Y2: int] :
                ( ( member_int @ Y2 @ ( B3 @ X2 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% ball_UN
thf(fact_1018_ball__UN,axiom,
    ! [B3: int > set_int,A3: set_int,P: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int @ X2 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A3 ) ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ A3 )
           => ! [Y2: int] :
                ( ( member_int @ Y2 @ ( B3 @ X2 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% ball_UN
thf(fact_1019_ball__UN,axiom,
    ! [B3: nat > set_nat,A3: set_nat,P: nat > $o] :
      ( ( ! [X2: nat] :
            ( ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
           => ! [Y2: nat] :
                ( ( member_nat @ Y2 @ ( B3 @ X2 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% ball_UN
thf(fact_1020_ball__UN,axiom,
    ! [B3: int > set_set_int,A3: set_int,P: set_int > $o] :
      ( ( ! [X2: set_int] :
            ( ( member_set_int @ X2 @ ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ B3 @ A3 ) ) )
           => ( P @ X2 ) ) )
      = ( ! [X2: int] :
            ( ( member_int @ X2 @ A3 )
           => ! [Y2: set_int] :
                ( ( member_set_int @ Y2 @ ( B3 @ X2 ) )
               => ( P @ Y2 ) ) ) ) ) ).

% ball_UN
thf(fact_1021_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_1022_UN__I,axiom,
    ! [A: nat,A3: set_nat,B: int,B3: nat > set_int] :
      ( ( member_nat @ A @ A3 )
     => ( ( member_int @ B @ ( B3 @ A ) )
       => ( member_int @ B @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B3 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1023_UN__I,axiom,
    ! [A: int,A3: set_int,B: int,B3: int > set_int] :
      ( ( member_int @ A @ A3 )
     => ( ( member_int @ B @ ( B3 @ A ) )
       => ( member_int @ B @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1024_UN__I,axiom,
    ! [A: nat,A3: set_nat,B: nat,B3: nat > set_nat] :
      ( ( member_nat @ A @ A3 )
     => ( ( member_nat @ B @ ( B3 @ A ) )
       => ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1025_UN__I,axiom,
    ! [A: int,A3: set_int,B: nat,B3: int > set_nat] :
      ( ( member_int @ A @ A3 )
     => ( ( member_nat @ B @ ( B3 @ A ) )
       => ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B3 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1026_UN__I,axiom,
    ! [A: set_int,A3: set_set_int,B: int,B3: set_int > set_int] :
      ( ( member_set_int @ A @ A3 )
     => ( ( member_int @ B @ ( B3 @ A ) )
       => ( member_int @ B @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ B3 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1027_UN__I,axiom,
    ! [A: set_int,A3: set_set_int,B: nat,B3: set_int > set_nat] :
      ( ( member_set_int @ A @ A3 )
     => ( ( member_nat @ B @ ( B3 @ A ) )
       => ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ B3 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1028_UN__I,axiom,
    ! [A: nat,A3: set_nat,B: set_int,B3: nat > set_set_int] :
      ( ( member_nat @ A @ A3 )
     => ( ( member_set_int @ B @ ( B3 @ A ) )
       => ( member_set_int @ B @ ( comple7281953568134767595et_int @ ( image_8927401050382224495et_int @ B3 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1029_UN__I,axiom,
    ! [A: int,A3: set_int,B: set_int,B3: int > set_set_int] :
      ( ( member_int @ A @ A3 )
     => ( ( member_set_int @ B @ ( B3 @ A ) )
       => ( member_set_int @ B @ ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ B3 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1030_UN__I,axiom,
    ! [A: nat,A3: set_nat,B: set_set_int,B3: nat > set_set_set_int] :
      ( ( member_nat @ A @ A3 )
     => ( ( member_set_set_int @ B @ ( B3 @ A ) )
       => ( member_set_set_int @ B @ ( comple1756060948637632417et_int @ ( image_924495994448699429et_int @ B3 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1031_UN__I,axiom,
    ! [A: int,A3: set_int,B: set_set_int,B3: int > set_set_set_int] :
      ( ( member_int @ A @ A3 )
     => ( ( member_set_set_int @ B @ ( B3 @ A ) )
       => ( member_set_set_int @ B @ ( comple1756060948637632417et_int @ ( image_63215931098211969et_int @ B3 @ A3 ) ) ) ) ) ).

% UN_I
thf(fact_1032_UN__iff,axiom,
    ! [B: int,B3: set_int > set_int,A3: set_set_int] :
      ( ( member_int @ B @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ B3 @ A3 ) ) )
      = ( ? [X2: set_int] :
            ( ( member_set_int @ X2 @ A3 )
            & ( member_int @ B @ ( B3 @ X2 ) ) ) ) ) ).

% UN_iff
thf(fact_1033_UN__iff,axiom,
    ! [B: int,B3: int > set_int,A3: set_int] :
      ( ( member_int @ B @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A3 ) ) )
      = ( ? [X2: int] :
            ( ( member_int @ X2 @ A3 )
            & ( member_int @ B @ ( B3 @ X2 ) ) ) ) ) ).

% UN_iff
thf(fact_1034_UN__iff,axiom,
    ! [B: nat,B3: nat > set_nat,A3: set_nat] :
      ( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) )
      = ( ? [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
            & ( member_nat @ B @ ( B3 @ X2 ) ) ) ) ) ).

% UN_iff
thf(fact_1035_UN__iff,axiom,
    ! [B: set_int,B3: int > set_set_int,A3: set_int] :
      ( ( member_set_int @ B @ ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ B3 @ A3 ) ) )
      = ( ? [X2: int] :
            ( ( member_int @ X2 @ A3 )
            & ( member_set_int @ B @ ( B3 @ X2 ) ) ) ) ) ).

% UN_iff
thf(fact_1036_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_1037_i_Oadd_Oinj__on__multc,axiom,
    ! [C: int] :
      ( ( 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 ) ) ) ) )
     => ( inj_on_int_int
        @ ^ [X2: 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 ) ) ) @ X2 @ 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 ) ) ) ) ) ) ).

% i.add.inj_on_multc
thf(fact_1038_i_Oadd_Oinj__on__cmult,axiom,
    ! [C: int] :
      ( ( 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 ) ) ) ) )
     => ( inj_on_int_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 ) @ ( 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.inj_on_cmult
thf(fact_1039_UN__constant,axiom,
    ! [A3: set_int,C: set_int] :
      ( ( ( A3 = bot_bot_set_int )
       => ( ( comple3221217463730067765et_int
            @ ( image_int_set_int
              @ ^ [Y2: int] : C
              @ A3 ) )
          = bot_bot_set_int ) )
      & ( ( A3 != bot_bot_set_int )
       => ( ( comple3221217463730067765et_int
            @ ( image_int_set_int
              @ ^ [Y2: int] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_1040_UN__constant,axiom,
    ! [A3: set_nat,C: set_int] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple3221217463730067765et_int
            @ ( image_nat_set_int
              @ ^ [Y2: nat] : C
              @ A3 ) )
          = bot_bot_set_int ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple3221217463730067765et_int
            @ ( image_nat_set_int
              @ ^ [Y2: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_1041_UN__constant,axiom,
    ! [A3: set_set_int,C: set_int] :
      ( ( ( A3 = bot_bot_set_set_int )
       => ( ( comple3221217463730067765et_int
            @ ( image_524474410958335435et_int
              @ ^ [Y2: set_int] : C
              @ A3 ) )
          = bot_bot_set_int ) )
      & ( ( A3 != bot_bot_set_set_int )
       => ( ( comple3221217463730067765et_int
            @ ( image_524474410958335435et_int
              @ ^ [Y2: set_int] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_1042_UN__constant,axiom,
    ! [A3: set_int,C: set_nat] :
      ( ( ( A3 = bot_bot_set_int )
       => ( ( comple7399068483239264473et_nat
            @ ( image_int_set_nat
              @ ^ [Y2: int] : C
              @ A3 ) )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_int )
       => ( ( comple7399068483239264473et_nat
            @ ( image_int_set_nat
              @ ^ [Y2: int] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_1043_UN__constant,axiom,
    ! [A3: set_nat,C: set_nat] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [Y2: nat] : C
              @ A3 ) )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple7399068483239264473et_nat
            @ ( image_nat_set_nat
              @ ^ [Y2: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_1044_UN__constant,axiom,
    ! [A3: set_set_int,C: set_nat] :
      ( ( ( A3 = bot_bot_set_set_int )
       => ( ( comple7399068483239264473et_nat
            @ ( image_4702325430467532143et_nat
              @ ^ [Y2: set_int] : C
              @ A3 ) )
          = bot_bot_set_nat ) )
      & ( ( A3 != bot_bot_set_set_int )
       => ( ( comple7399068483239264473et_nat
            @ ( image_4702325430467532143et_nat
              @ ^ [Y2: set_int] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_1045_UN__constant,axiom,
    ! [A3: set_int,C: set_set_int] :
      ( ( ( A3 = bot_bot_set_int )
       => ( ( comple7281953568134767595et_int
            @ ( image_6617583118289273547et_int
              @ ^ [Y2: int] : C
              @ A3 ) )
          = bot_bot_set_set_int ) )
      & ( ( A3 != bot_bot_set_int )
       => ( ( comple7281953568134767595et_int
            @ ( image_6617583118289273547et_int
              @ ^ [Y2: int] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_1046_UN__constant,axiom,
    ! [A3: set_nat,C: set_set_int] :
      ( ( ( A3 = bot_bot_set_nat )
       => ( ( comple7281953568134767595et_int
            @ ( image_8927401050382224495et_int
              @ ^ [Y2: nat] : C
              @ A3 ) )
          = bot_bot_set_set_int ) )
      & ( ( A3 != bot_bot_set_nat )
       => ( ( comple7281953568134767595et_int
            @ ( image_8927401050382224495et_int
              @ ^ [Y2: nat] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_1047_UN__constant,axiom,
    ! [A3: set_set_int,C: set_set_int] :
      ( ( ( A3 = bot_bot_set_set_int )
       => ( ( comple7281953568134767595et_int
            @ ( image_1010086626112315521et_int
              @ ^ [Y2: set_int] : C
              @ A3 ) )
          = bot_bot_set_set_int ) )
      & ( ( A3 != bot_bot_set_set_int )
       => ( ( comple7281953568134767595et_int
            @ ( image_1010086626112315521et_int
              @ ^ [Y2: set_int] : C
              @ A3 ) )
          = C ) ) ) ).

% UN_constant
thf(fact_1048_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_1049_image__strict__mono,axiom,
    ! [F: nat > set_nat,B3: set_nat,A3: set_nat] :
      ( ( inj_on_nat_set_nat @ F @ B3 )
     => ( ( ord_less_set_nat @ A3 @ B3 )
       => ( ord_less_set_set_nat @ ( image_nat_set_nat @ F @ A3 ) @ ( image_nat_set_nat @ F @ B3 ) ) ) ) ).

% image_strict_mono
thf(fact_1050_image__strict__mono,axiom,
    ! [F: int > set_set_int,B3: set_int,A3: set_int] :
      ( ( inj_on3305102505822124087et_int @ F @ B3 )
     => ( ( ord_less_set_int @ A3 @ B3 )
       => ( ord_le4562804192517611682et_int @ ( image_6617583118289273547et_int @ F @ A3 ) @ ( image_6617583118289273547et_int @ F @ B3 ) ) ) ) ).

% image_strict_mono
thf(fact_1051_image__strict__mono,axiom,
    ! [F: int > int,B3: set_int,A3: set_int] :
      ( ( inj_on_int_int @ F @ B3 )
     => ( ( ord_less_set_int @ A3 @ B3 )
       => ( ord_less_set_int @ ( image_int_int @ F @ A3 ) @ ( image_int_int @ F @ B3 ) ) ) ) ).

% image_strict_mono
thf(fact_1052_image__strict__mono,axiom,
    ! [F: set_int > set_int,B3: set_set_int,A3: set_set_int] :
      ( ( inj_on6435365835345961783et_int @ F @ B3 )
     => ( ( ord_less_set_set_int @ A3 @ B3 )
       => ( ord_less_set_set_int @ ( image_524474410958335435et_int @ F @ A3 ) @ ( image_524474410958335435et_int @ F @ B3 ) ) ) ) ).

% image_strict_mono
thf(fact_1053_image__strict__mono,axiom,
    ! [F: int > set_int,B3: set_int,A3: set_int] :
      ( ( inj_on_int_set_int @ F @ B3 )
     => ( ( ord_less_set_int @ A3 @ B3 )
       => ( ord_less_set_set_int @ ( image_int_set_int @ F @ A3 ) @ ( image_int_set_int @ F @ B3 ) ) ) ) ).

% image_strict_mono
thf(fact_1054_inj__on__UNION__chain,axiom,
    ! [I5: set_nat,A3: nat > set_int,F: int > int] :
      ( ! [I3: nat,J3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ( member_nat @ J3 @ I5 )
           => ( ( ord_less_eq_set_int @ ( A3 @ I3 ) @ ( A3 @ J3 ) )
              | ( ord_less_eq_set_int @ ( A3 @ J3 ) @ ( A3 @ I3 ) ) ) ) )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( inj_on_int_int @ F @ ( A3 @ I3 ) ) )
       => ( inj_on_int_int @ F @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ A3 @ I5 ) ) ) ) ) ).

% inj_on_UNION_chain
thf(fact_1055_inj__on__UNION__chain,axiom,
    ! [I5: set_int,A3: int > set_int,F: int > int] :
      ( ! [I3: int,J3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ( member_int @ J3 @ I5 )
           => ( ( ord_less_eq_set_int @ ( A3 @ I3 ) @ ( A3 @ J3 ) )
              | ( ord_less_eq_set_int @ ( A3 @ J3 ) @ ( A3 @ I3 ) ) ) ) )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( inj_on_int_int @ F @ ( A3 @ I3 ) ) )
       => ( inj_on_int_int @ F @ ( comple3221217463730067765et_int @ ( image_int_set_int @ A3 @ I5 ) ) ) ) ) ).

% inj_on_UNION_chain
thf(fact_1056_inj__on__UNION__chain,axiom,
    ! [I5: set_set_int,A3: set_int > set_int,F: int > int] :
      ( ! [I3: set_int,J3: set_int] :
          ( ( member_set_int @ I3 @ I5 )
         => ( ( member_set_int @ J3 @ I5 )
           => ( ( ord_less_eq_set_int @ ( A3 @ I3 ) @ ( A3 @ J3 ) )
              | ( ord_less_eq_set_int @ ( A3 @ J3 ) @ ( A3 @ I3 ) ) ) ) )
     => ( ! [I3: set_int] :
            ( ( member_set_int @ I3 @ I5 )
           => ( inj_on_int_int @ F @ ( A3 @ I3 ) ) )
       => ( inj_on_int_int @ F @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ A3 @ I5 ) ) ) ) ) ).

% inj_on_UNION_chain
thf(fact_1057_inj__on__UNION__chain,axiom,
    ! [I5: set_nat,A3: nat > set_int,F: int > set_int] :
      ( ! [I3: nat,J3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ( member_nat @ J3 @ I5 )
           => ( ( ord_less_eq_set_int @ ( A3 @ I3 ) @ ( A3 @ J3 ) )
              | ( ord_less_eq_set_int @ ( A3 @ J3 ) @ ( A3 @ I3 ) ) ) ) )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( inj_on_int_set_int @ F @ ( A3 @ I3 ) ) )
       => ( inj_on_int_set_int @ F @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ A3 @ I5 ) ) ) ) ) ).

% inj_on_UNION_chain
thf(fact_1058_inj__on__UNION__chain,axiom,
    ! [I5: set_int,A3: int > set_int,F: int > set_int] :
      ( ! [I3: int,J3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ( member_int @ J3 @ I5 )
           => ( ( ord_less_eq_set_int @ ( A3 @ I3 ) @ ( A3 @ J3 ) )
              | ( ord_less_eq_set_int @ ( A3 @ J3 ) @ ( A3 @ I3 ) ) ) ) )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( inj_on_int_set_int @ F @ ( A3 @ I3 ) ) )
       => ( inj_on_int_set_int @ F @ ( comple3221217463730067765et_int @ ( image_int_set_int @ A3 @ I5 ) ) ) ) ) ).

% inj_on_UNION_chain
thf(fact_1059_inj__on__UNION__chain,axiom,
    ! [I5: set_set_set_int,A3: set_set_int > set_int,F: int > int] :
      ( ! [I3: set_set_int,J3: set_set_int] :
          ( ( member_set_set_int @ I3 @ I5 )
         => ( ( member_set_set_int @ J3 @ I5 )
           => ( ( ord_less_eq_set_int @ ( A3 @ I3 ) @ ( A3 @ J3 ) )
              | ( ord_less_eq_set_int @ ( A3 @ J3 ) @ ( A3 @ I3 ) ) ) ) )
     => ( ! [I3: set_set_int] :
            ( ( member_set_set_int @ I3 @ I5 )
           => ( inj_on_int_int @ F @ ( A3 @ I3 ) ) )
       => ( inj_on_int_int @ F @ ( comple3221217463730067765et_int @ ( image_3513010637850279041et_int @ A3 @ I5 ) ) ) ) ) ).

% inj_on_UNION_chain
thf(fact_1060_inj__on__UNION__chain,axiom,
    ! [I5: set_set_int,A3: set_int > set_int,F: int > set_int] :
      ( ! [I3: set_int,J3: set_int] :
          ( ( member_set_int @ I3 @ I5 )
         => ( ( member_set_int @ J3 @ I5 )
           => ( ( ord_less_eq_set_int @ ( A3 @ I3 ) @ ( A3 @ J3 ) )
              | ( ord_less_eq_set_int @ ( A3 @ J3 ) @ ( A3 @ I3 ) ) ) ) )
     => ( ! [I3: set_int] :
            ( ( member_set_int @ I3 @ I5 )
           => ( inj_on_int_set_int @ F @ ( A3 @ I3 ) ) )
       => ( inj_on_int_set_int @ F @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ A3 @ I5 ) ) ) ) ) ).

% inj_on_UNION_chain
thf(fact_1061_inj__on__UNION__chain,axiom,
    ! [I5: set_nat,A3: nat > set_set_int,F: set_int > set_int] :
      ( ! [I3: nat,J3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ( member_nat @ J3 @ I5 )
           => ( ( ord_le4403425263959731960et_int @ ( A3 @ I3 ) @ ( A3 @ J3 ) )
              | ( ord_le4403425263959731960et_int @ ( A3 @ J3 ) @ ( A3 @ I3 ) ) ) ) )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( inj_on6435365835345961783et_int @ F @ ( A3 @ I3 ) ) )
       => ( inj_on6435365835345961783et_int @ F @ ( comple7281953568134767595et_int @ ( image_8927401050382224495et_int @ A3 @ I5 ) ) ) ) ) ).

% inj_on_UNION_chain
thf(fact_1062_inj__on__UNION__chain,axiom,
    ! [I5: set_int,A3: int > set_set_int,F: set_int > set_int] :
      ( ! [I3: int,J3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ( member_int @ J3 @ I5 )
           => ( ( ord_le4403425263959731960et_int @ ( A3 @ I3 ) @ ( A3 @ J3 ) )
              | ( ord_le4403425263959731960et_int @ ( A3 @ J3 ) @ ( A3 @ I3 ) ) ) ) )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( inj_on6435365835345961783et_int @ F @ ( A3 @ I3 ) ) )
       => ( inj_on6435365835345961783et_int @ F @ ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ A3 @ I5 ) ) ) ) ) ).

% inj_on_UNION_chain
thf(fact_1063_inj__on__UNION__chain,axiom,
    ! [I5: set_set_set_int,A3: set_set_int > set_int,F: int > set_int] :
      ( ! [I3: set_set_int,J3: set_set_int] :
          ( ( member_set_set_int @ I3 @ I5 )
         => ( ( member_set_set_int @ J3 @ I5 )
           => ( ( ord_less_eq_set_int @ ( A3 @ I3 ) @ ( A3 @ J3 ) )
              | ( ord_less_eq_set_int @ ( A3 @ J3 ) @ ( A3 @ I3 ) ) ) ) )
     => ( ! [I3: set_set_int] :
            ( ( member_set_set_int @ I3 @ I5 )
           => ( inj_on_int_set_int @ F @ ( A3 @ I3 ) ) )
       => ( inj_on_int_set_int @ F @ ( comple3221217463730067765et_int @ ( image_3513010637850279041et_int @ A3 @ I5 ) ) ) ) ) ).

% inj_on_UNION_chain
thf(fact_1064_inj__on__image__mem__iff,axiom,
    ! [F: nat > nat,B3: set_nat,A: nat,A3: set_nat] :
      ( ( inj_on_nat_nat @ F @ B3 )
     => ( ( member_nat @ A @ B3 )
       => ( ( ord_less_eq_set_nat @ A3 @ B3 )
         => ( ( member_nat @ ( F @ A ) @ ( image_nat_nat @ F @ A3 ) )
            = ( member_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1065_inj__on__image__mem__iff,axiom,
    ! [F: nat > int,B3: set_nat,A: nat,A3: set_nat] :
      ( ( inj_on_nat_int @ F @ B3 )
     => ( ( member_nat @ A @ B3 )
       => ( ( ord_less_eq_set_nat @ A3 @ B3 )
         => ( ( member_int @ ( F @ A ) @ ( image_nat_int @ F @ A3 ) )
            = ( member_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1066_inj__on__image__mem__iff,axiom,
    ! [F: int > nat,B3: set_int,A: int,A3: set_int] :
      ( ( inj_on_int_nat @ F @ B3 )
     => ( ( member_int @ A @ B3 )
       => ( ( ord_less_eq_set_int @ A3 @ B3 )
         => ( ( member_nat @ ( F @ A ) @ ( image_int_nat @ F @ A3 ) )
            = ( member_int @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1067_inj__on__image__mem__iff,axiom,
    ! [F: int > int,B3: set_int,A: int,A3: set_int] :
      ( ( inj_on_int_int @ F @ B3 )
     => ( ( member_int @ A @ B3 )
       => ( ( ord_less_eq_set_int @ A3 @ B3 )
         => ( ( member_int @ ( F @ A ) @ ( image_int_int @ F @ A3 ) )
            = ( member_int @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1068_inj__on__image__mem__iff,axiom,
    ! [F: nat > set_nat,B3: set_nat,A: nat,A3: set_nat] :
      ( ( inj_on_nat_set_nat @ F @ B3 )
     => ( ( member_nat @ A @ B3 )
       => ( ( ord_less_eq_set_nat @ A3 @ B3 )
         => ( ( member_set_nat @ ( F @ A ) @ ( image_nat_set_nat @ F @ A3 ) )
            = ( member_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1069_inj__on__image__mem__iff,axiom,
    ! [F: nat > set_int,B3: set_nat,A: nat,A3: set_nat] :
      ( ( inj_on_nat_set_int @ F @ B3 )
     => ( ( member_nat @ A @ B3 )
       => ( ( ord_less_eq_set_nat @ A3 @ B3 )
         => ( ( member_set_int @ ( F @ A ) @ ( image_nat_set_int @ F @ A3 ) )
            = ( member_nat @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1070_inj__on__image__mem__iff,axiom,
    ! [F: int > set_int,B3: set_int,A: int,A3: set_int] :
      ( ( inj_on_int_set_int @ F @ B3 )
     => ( ( member_int @ A @ B3 )
       => ( ( ord_less_eq_set_int @ A3 @ B3 )
         => ( ( member_set_int @ ( F @ A ) @ ( image_int_set_int @ F @ A3 ) )
            = ( member_int @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1071_inj__on__image__mem__iff,axiom,
    ! [F: set_int > nat,B3: set_set_int,A: set_int,A3: set_set_int] :
      ( ( inj_on_set_int_nat @ F @ B3 )
     => ( ( member_set_int @ A @ B3 )
       => ( ( ord_le4403425263959731960et_int @ A3 @ B3 )
         => ( ( member_nat @ ( F @ A ) @ ( image_set_int_nat @ F @ A3 ) )
            = ( member_set_int @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1072_inj__on__image__mem__iff,axiom,
    ! [F: set_int > int,B3: set_set_int,A: set_int,A3: set_set_int] :
      ( ( inj_on_set_int_int @ F @ B3 )
     => ( ( member_set_int @ A @ B3 )
       => ( ( ord_le4403425263959731960et_int @ A3 @ B3 )
         => ( ( member_int @ ( F @ A ) @ ( image_set_int_int @ F @ A3 ) )
            = ( member_set_int @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1073_inj__on__image__mem__iff,axiom,
    ! [F: set_set_int > nat,B3: set_set_set_int,A: set_set_int,A3: set_set_set_int] :
      ( ( inj_on1361139300194298331nt_nat @ F @ B3 )
     => ( ( member_set_set_int @ A @ B3 )
       => ( ( ord_le4317611570275147438et_int @ A3 @ B3 )
         => ( ( member_nat @ ( F @ A ) @ ( image_4673619912661447791nt_nat @ F @ A3 ) )
            = ( member_set_set_int @ A @ A3 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1074_inj__on__image__eq__iff,axiom,
    ! [F: nat > set_nat,C2: set_nat,A3: set_nat,B3: set_nat] :
      ( ( inj_on_nat_set_nat @ F @ C2 )
     => ( ( ord_less_eq_set_nat @ A3 @ C2 )
       => ( ( ord_less_eq_set_nat @ B3 @ C2 )
         => ( ( ( image_nat_set_nat @ F @ A3 )
              = ( image_nat_set_nat @ F @ B3 ) )
            = ( A3 = B3 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_1075_inj__on__image__eq__iff,axiom,
    ! [F: int > set_set_int,C2: set_int,A3: set_int,B3: set_int] :
      ( ( inj_on3305102505822124087et_int @ F @ C2 )
     => ( ( ord_less_eq_set_int @ A3 @ C2 )
       => ( ( ord_less_eq_set_int @ B3 @ C2 )
         => ( ( ( image_6617583118289273547et_int @ F @ A3 )
              = ( image_6617583118289273547et_int @ F @ B3 ) )
            = ( A3 = B3 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_1076_inj__on__image__eq__iff,axiom,
    ! [F: int > int,C2: set_int,A3: set_int,B3: set_int] :
      ( ( inj_on_int_int @ F @ C2 )
     => ( ( ord_less_eq_set_int @ A3 @ C2 )
       => ( ( ord_less_eq_set_int @ B3 @ C2 )
         => ( ( ( image_int_int @ F @ A3 )
              = ( image_int_int @ F @ B3 ) )
            = ( A3 = B3 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_1077_inj__on__image__eq__iff,axiom,
    ! [F: int > set_int,C2: set_int,A3: set_int,B3: set_int] :
      ( ( inj_on_int_set_int @ F @ C2 )
     => ( ( ord_less_eq_set_int @ A3 @ C2 )
       => ( ( ord_less_eq_set_int @ B3 @ C2 )
         => ( ( ( image_int_set_int @ F @ A3 )
              = ( image_int_set_int @ F @ B3 ) )
            = ( A3 = B3 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_1078_inj__on__image__eq__iff,axiom,
    ! [F: set_int > set_int,C2: set_set_int,A3: set_set_int,B3: set_set_int] :
      ( ( inj_on6435365835345961783et_int @ F @ C2 )
     => ( ( ord_le4403425263959731960et_int @ A3 @ C2 )
       => ( ( ord_le4403425263959731960et_int @ B3 @ C2 )
         => ( ( ( image_524474410958335435et_int @ F @ A3 )
              = ( image_524474410958335435et_int @ F @ B3 ) )
            = ( A3 = B3 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_1079_subset__inj__on,axiom,
    ! [F: int > int,B3: set_int,A3: set_int] :
      ( ( inj_on_int_int @ F @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( inj_on_int_int @ F @ A3 ) ) ) ).

% subset_inj_on
thf(fact_1080_subset__inj__on,axiom,
    ! [F: int > set_int,B3: set_int,A3: set_int] :
      ( ( inj_on_int_set_int @ F @ B3 )
     => ( ( ord_less_eq_set_int @ A3 @ B3 )
       => ( inj_on_int_set_int @ F @ A3 ) ) ) ).

% subset_inj_on
thf(fact_1081_subset__inj__on,axiom,
    ! [F: set_int > set_int,B3: set_set_int,A3: set_set_int] :
      ( ( inj_on6435365835345961783et_int @ F @ B3 )
     => ( ( ord_le4403425263959731960et_int @ A3 @ B3 )
       => ( inj_on6435365835345961783et_int @ F @ A3 ) ) ) ).

% subset_inj_on
thf(fact_1082_inj__on__subset,axiom,
    ! [F: int > int,A3: set_int,B3: set_int] :
      ( ( inj_on_int_int @ F @ A3 )
     => ( ( ord_less_eq_set_int @ B3 @ A3 )
       => ( inj_on_int_int @ F @ B3 ) ) ) ).

% inj_on_subset
thf(fact_1083_inj__on__subset,axiom,
    ! [F: int > set_int,A3: set_int,B3: set_int] :
      ( ( inj_on_int_set_int @ F @ A3 )
     => ( ( ord_less_eq_set_int @ B3 @ A3 )
       => ( inj_on_int_set_int @ F @ B3 ) ) ) ).

% inj_on_subset
thf(fact_1084_inj__on__subset,axiom,
    ! [F: set_int > set_int,A3: set_set_int,B3: set_set_int] :
      ( ( inj_on6435365835345961783et_int @ F @ A3 )
     => ( ( ord_le4403425263959731960et_int @ B3 @ A3 )
       => ( inj_on6435365835345961783et_int @ F @ B3 ) ) ) ).

% inj_on_subset
thf(fact_1085_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_1086_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_1087_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_1088_injI,axiom,
    ! [F: set_int > set_int] :
      ( ! [X5: set_int,Y4: set_int] :
          ( ( ( F @ X5 )
            = ( F @ Y4 ) )
         => ( X5 = Y4 ) )
     => ( inj_on6435365835345961783et_int @ F @ top_top_set_set_int ) ) ).

% injI
thf(fact_1089_injI,axiom,
    ! [F: int > int] :
      ( ! [X5: int,Y4: int] :
          ( ( ( F @ X5 )
            = ( F @ Y4 ) )
         => ( X5 = Y4 ) )
     => ( inj_on_int_int @ F @ top_top_set_int ) ) ).

% injI
thf(fact_1090_injI,axiom,
    ! [F: int > set_int] :
      ( ! [X5: int,Y4: int] :
          ( ( ( F @ X5 )
            = ( F @ Y4 ) )
         => ( X5 = Y4 ) )
     => ( inj_on_int_set_int @ F @ top_top_set_int ) ) ).

% injI
thf(fact_1091_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_1092_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_1093_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_1094_inj__def,axiom,
    ! [F: set_int > set_int] :
      ( ( inj_on6435365835345961783et_int @ F @ top_top_set_set_int )
      = ( ! [X2: set_int,Y2: set_int] :
            ( ( ( F @ X2 )
              = ( F @ Y2 ) )
           => ( X2 = Y2 ) ) ) ) ).

% inj_def
thf(fact_1095_inj__def,axiom,
    ! [F: int > int] :
      ( ( inj_on_int_int @ F @ top_top_set_int )
      = ( ! [X2: int,Y2: int] :
            ( ( ( F @ X2 )
              = ( F @ Y2 ) )
           => ( X2 = Y2 ) ) ) ) ).

% inj_def
thf(fact_1096_inj__def,axiom,
    ! [F: int > set_int] :
      ( ( inj_on_int_set_int @ F @ top_top_set_int )
      = ( ! [X2: int,Y2: int] :
            ( ( ( F @ X2 )
              = ( F @ Y2 ) )
           => ( X2 = Y2 ) ) ) ) ).

% inj_def
thf(fact_1097_linorder__inj__onI_H,axiom,
    ! [A3: set_int,F: int > int] :
      ( ! [I3: int,J3: int] :
          ( ( member_int @ I3 @ A3 )
         => ( ( member_int @ J3 @ A3 )
           => ( ( ord_less_int @ I3 @ J3 )
             => ( ( F @ I3 )
               != ( F @ J3 ) ) ) ) )
     => ( inj_on_int_int @ F @ A3 ) ) ).

% linorder_inj_onI'
thf(fact_1098_linorder__inj__onI_H,axiom,
    ! [A3: set_int,F: int > set_int] :
      ( ! [I3: int,J3: int] :
          ( ( member_int @ I3 @ A3 )
         => ( ( member_int @ J3 @ A3 )
           => ( ( ord_less_int @ I3 @ J3 )
             => ( ( F @ I3 )
               != ( F @ J3 ) ) ) ) )
     => ( inj_on_int_set_int @ F @ A3 ) ) ).

% linorder_inj_onI'
thf(fact_1099_UN__extend__simps_I8_J,axiom,
    ! [B3: int > set_int,A3: set_set_int] :
      ( ( comple3221217463730067765et_int
        @ ( image_524474410958335435et_int
          @ ^ [Y2: set_int] : ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ Y2 ) )
          @ A3 ) )
      = ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ ( comple3221217463730067765et_int @ A3 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_1100_UN__extend__simps_I8_J,axiom,
    ! [B3: nat > set_int,A3: set_set_nat] :
      ( ( comple3221217463730067765et_int
        @ ( image_3739036796817536367et_int
          @ ^ [Y2: set_nat] : ( comple3221217463730067765et_int @ ( image_nat_set_int @ B3 @ Y2 ) )
          @ A3 ) )
      = ( comple3221217463730067765et_int @ ( image_nat_set_int @ B3 @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_1101_UN__extend__simps_I8_J,axiom,
    ! [B3: set_int > set_int,A3: set_set_set_int] :
      ( ( comple3221217463730067765et_int
        @ ( image_3513010637850279041et_int
          @ ^ [Y2: set_set_int] : ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ B3 @ Y2 ) )
          @ A3 ) )
      = ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ B3 @ ( comple7281953568134767595et_int @ A3 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_1102_UN__extend__simps_I8_J,axiom,
    ! [B3: int > set_nat,A3: set_set_int] :
      ( ( comple7399068483239264473et_nat
        @ ( image_4702325430467532143et_nat
          @ ^ [Y2: set_int] : ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B3 @ Y2 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B3 @ ( comple3221217463730067765et_int @ A3 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_1103_UN__extend__simps_I8_J,axiom,
    ! [B3: nat > set_nat,A3: set_set_nat] :
      ( ( comple7399068483239264473et_nat
        @ ( image_7916887816326733075et_nat
          @ ^ [Y2: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ Y2 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_1104_UN__extend__simps_I8_J,axiom,
    ! [B3: set_int > set_nat,A3: set_set_set_int] :
      ( ( comple7399068483239264473et_nat
        @ ( image_7690861657359475749et_nat
          @ ^ [Y2: set_set_int] : ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ B3 @ Y2 ) )
          @ A3 ) )
      = ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ B3 @ ( comple7281953568134767595et_int @ A3 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_1105_UN__extend__simps_I8_J,axiom,
    ! [B3: int > set_set_int,A3: set_set_int] :
      ( ( comple7281953568134767595et_int
        @ ( image_1010086626112315521et_int
          @ ^ [Y2: set_int] : ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ B3 @ Y2 ) )
          @ A3 ) )
      = ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ B3 @ ( comple3221217463730067765et_int @ A3 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_1106_UN__extend__simps_I8_J,axiom,
    ! [B3: nat > set_set_int,A3: set_set_nat] :
      ( ( comple7281953568134767595et_int
        @ ( image_4234937972324292645et_int
          @ ^ [Y2: set_nat] : ( comple7281953568134767595et_int @ ( image_8927401050382224495et_int @ B3 @ Y2 ) )
          @ A3 ) )
      = ( comple7281953568134767595et_int @ ( image_8927401050382224495et_int @ B3 @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_1107_UN__extend__simps_I8_J,axiom,
    ! [B3: set_int > set_set_int,A3: set_set_set_int] :
      ( ( comple7281953568134767595et_int
        @ ( image_2718908958957770551et_int
          @ ^ [Y2: set_set_int] : ( comple7281953568134767595et_int @ ( image_1010086626112315521et_int @ B3 @ Y2 ) )
          @ A3 ) )
      = ( comple7281953568134767595et_int @ ( image_1010086626112315521et_int @ B3 @ ( comple7281953568134767595et_int @ A3 ) ) ) ) ).

% UN_extend_simps(8)
thf(fact_1108_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 ) )
        = ( ? [X2: int] :
              ( ( B
                = ( F @ X2 ) )
              & ! [Y2: int] :
                  ( ( B
                    = ( F @ Y2 ) )
                 => ( Y2 = X2 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1109_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 ) )
        = ( ? [X2: int] :
              ( ( B
                = ( F @ X2 ) )
              & ! [Y2: int] :
                  ( ( B
                    = ( F @ Y2 ) )
                 => ( Y2 = X2 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1110_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 ) )
        = ( ? [X2: nat] :
              ( ( B
                = ( F @ X2 ) )
              & ! [Y2: nat] :
                  ( ( B
                    = ( F @ Y2 ) )
                 => ( Y2 = X2 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1111_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 ) )
        = ( ? [X2: nat] :
              ( ( B
                = ( F @ X2 ) )
              & ! [Y2: nat] :
                  ( ( B
                    = ( F @ Y2 ) )
                 => ( Y2 = X2 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1112_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 ) )
        = ( ? [X2: int] :
              ( ( B
                = ( F @ X2 ) )
              & ! [Y2: int] :
                  ( ( B
                    = ( F @ Y2 ) )
                 => ( Y2 = X2 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1113_range__ex1__eq,axiom,
    ! [F: nat > set_nat,B: set_nat] :
      ( ( inj_on_nat_set_nat @ F @ top_top_set_nat )
     => ( ( member_set_nat @ B @ ( image_nat_set_nat @ F @ top_top_set_nat ) )
        = ( ? [X2: nat] :
              ( ( B
                = ( F @ X2 ) )
              & ! [Y2: nat] :
                  ( ( B
                    = ( F @ Y2 ) )
                 => ( Y2 = X2 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1114_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 ) )
        = ( ? [X2: nat] :
              ( ( B
                = ( F @ X2 ) )
              & ! [Y2: nat] :
                  ( ( B
                    = ( F @ Y2 ) )
                 => ( Y2 = X2 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1115_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 ) )
        = ( ? [X2: set_int] :
              ( ( B
                = ( F @ X2 ) )
              & ! [Y2: set_int] :
                  ( ( B
                    = ( F @ Y2 ) )
                 => ( Y2 = X2 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1116_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 ) )
        = ( ? [X2: int] :
              ( ( B
                = ( F @ X2 ) )
              & ! [Y2: int] :
                  ( ( B
                    = ( F @ Y2 ) )
                 => ( Y2 = X2 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1117_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 ) )
        = ( ? [X2: nat] :
              ( ( B
                = ( F @ X2 ) )
              & ! [Y2: nat] :
                  ( ( B
                    = ( F @ Y2 ) )
                 => ( Y2 = X2 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1118_inj__image__eq__iff,axiom,
    ! [F: set_int > set_int,A3: set_set_int,B3: set_set_int] :
      ( ( inj_on6435365835345961783et_int @ F @ top_top_set_set_int )
     => ( ( ( image_524474410958335435et_int @ F @ A3 )
          = ( image_524474410958335435et_int @ F @ B3 ) )
        = ( A3 = B3 ) ) ) ).

% inj_image_eq_iff
thf(fact_1119_inj__image__eq__iff,axiom,
    ! [F: int > set_set_int,A3: set_int,B3: set_int] :
      ( ( inj_on3305102505822124087et_int @ F @ top_top_set_int )
     => ( ( ( image_6617583118289273547et_int @ F @ A3 )
          = ( image_6617583118289273547et_int @ F @ B3 ) )
        = ( A3 = B3 ) ) ) ).

% inj_image_eq_iff
thf(fact_1120_inj__image__eq__iff,axiom,
    ! [F: int > int,A3: set_int,B3: set_int] :
      ( ( inj_on_int_int @ F @ top_top_set_int )
     => ( ( ( image_int_int @ F @ A3 )
          = ( image_int_int @ F @ B3 ) )
        = ( A3 = B3 ) ) ) ).

% inj_image_eq_iff
thf(fact_1121_inj__image__eq__iff,axiom,
    ! [F: int > set_int,A3: set_int,B3: set_int] :
      ( ( inj_on_int_set_int @ F @ top_top_set_int )
     => ( ( ( image_int_set_int @ F @ A3 )
          = ( image_int_set_int @ F @ B3 ) )
        = ( A3 = B3 ) ) ) ).

% inj_image_eq_iff
thf(fact_1122_inj__image__eq__iff,axiom,
    ! [F: nat > set_nat,A3: set_nat,B3: set_nat] :
      ( ( inj_on_nat_set_nat @ F @ top_top_set_nat )
     => ( ( ( image_nat_set_nat @ F @ A3 )
          = ( image_nat_set_nat @ F @ B3 ) )
        = ( A3 = B3 ) ) ) ).

% inj_image_eq_iff
thf(fact_1123_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_1124_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_1125_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_1126_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_1127_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_1128_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_1129_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_1130_inj__image__mem__iff,axiom,
    ! [F: nat > set_nat,A: nat,A3: set_nat] :
      ( ( inj_on_nat_set_nat @ F @ top_top_set_nat )
     => ( ( member_set_nat @ ( F @ A ) @ ( image_nat_set_nat @ F @ A3 ) )
        = ( member_nat @ A @ A3 ) ) ) ).

% inj_image_mem_iff
thf(fact_1131_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_1132_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_1133_Sup__set__def,axiom,
    ( comple1756060948637632417et_int
    = ( ^ [A6: set_set_set_set_int] :
          ( collect_set_set_int
          @ ^ [X2: set_set_int] : ( complete_Sup_Sup_o @ ( image_2622566851371609091_int_o @ ( member_set_set_int @ X2 ) @ A6 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1134_Sup__set__def,axiom,
    ( comple8192055500899288118et_int
    = ( ^ [A6: set_set_int_set_int] :
          ( collect_int_set_int
          @ ^ [X2: int > set_int] : ( complete_Sup_Sup_o @ ( image_5276478557982445742_int_o @ ( member_int_set_int @ X2 ) @ A6 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1135_Sup__set__def,axiom,
    ( comple3221217463730067765et_int
    = ( ^ [A6: set_set_int] :
          ( collect_int
          @ ^ [X2: int] : ( complete_Sup_Sup_o @ ( image_set_int_o @ ( member_int @ X2 ) @ A6 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1136_Sup__set__def,axiom,
    ( comple7399068483239264473et_nat
    = ( ^ [A6: set_set_nat] :
          ( collect_nat
          @ ^ [X2: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X2 ) @ A6 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1137_Sup__set__def,axiom,
    ( comple7281953568134767595et_int
    = ( ^ [A6: set_set_set_int] :
          ( collect_set_int
          @ ^ [X2: set_int] : ( complete_Sup_Sup_o @ ( image_set_set_int_o @ ( member_set_int @ X2 ) @ A6 ) ) ) ) ) ).

% Sup_set_def
thf(fact_1138_inj__on__image,axiom,
    ! [F: int > set_set_int,A3: set_set_int] :
      ( ( inj_on3305102505822124087et_int @ F @ ( comple3221217463730067765et_int @ A3 ) )
     => ( inj_on5116562847525634467et_int @ ( image_6617583118289273547et_int @ F ) @ A3 ) ) ).

% inj_on_image
thf(fact_1139_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_1140_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_1141_inj__on__image,axiom,
    ! [F: nat > set_nat,A3: set_set_nat] :
      ( ( inj_on_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ A3 ) )
     => ( inj_on2776966659131765557et_nat @ ( image_nat_set_nat @ F ) @ A3 ) ) ).

% inj_on_image
thf(fact_1142_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_1143_inj__on__image__iff,axiom,
    ! [A3: set_int,G: int > int,F: int > int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A3 )
         => ! [Xa: int] :
              ( ( member_int @ Xa @ A3 )
             => ( ( ( G @ ( F @ X5 ) )
                  = ( G @ ( F @ Xa ) ) )
                = ( ( G @ X5 )
                  = ( 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_1144_inj__on__image__iff,axiom,
    ! [A3: set_int,G: int > set_int,F: int > int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A3 )
         => ! [Xa: int] :
              ( ( member_int @ Xa @ A3 )
             => ( ( ( G @ ( F @ X5 ) )
                  = ( G @ ( F @ Xa ) ) )
                = ( ( G @ X5 )
                  = ( 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_1145_inj__on__image__iff,axiom,
    ! [A3: set_set_int,G: set_int > set_int,F: set_int > set_int] :
      ( ! [X5: set_int] :
          ( ( member_set_int @ X5 @ A3 )
         => ! [Xa: set_int] :
              ( ( member_set_int @ Xa @ A3 )
             => ( ( ( G @ ( F @ X5 ) )
                  = ( G @ ( F @ Xa ) ) )
                = ( ( G @ X5 )
                  = ( 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_1146_UN__UN__flatten,axiom,
    ! [C2: int > set_int,B3: int > set_int,A3: set_int] :
      ( ( comple3221217463730067765et_int @ ( image_int_set_int @ C2 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A3 ) ) ) )
      = ( comple3221217463730067765et_int
        @ ( image_int_set_int
          @ ^ [Y2: int] : ( comple3221217463730067765et_int @ ( image_int_set_int @ C2 @ ( B3 @ Y2 ) ) )
          @ A3 ) ) ) ).

% UN_UN_flatten
thf(fact_1147_UN__UN__flatten,axiom,
    ! [C2: nat > set_int,B3: int > set_nat,A3: set_int] :
      ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ C2 @ ( comple7399068483239264473et_nat @ ( image_int_set_nat @ B3 @ A3 ) ) ) )
      = ( comple3221217463730067765et_int
        @ ( image_int_set_int
          @ ^ [Y2: int] : ( comple3221217463730067765et_int @ ( image_nat_set_int @ C2 @ ( B3 @ Y2 ) ) )
          @ A3 ) ) ) ).

% UN_UN_flatten
thf(fact_1148_UN__UN__flatten,axiom,
    ! [C2: nat > set_int,B3: nat > set_nat,A3: set_nat] :
      ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ C2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) )
      = ( comple3221217463730067765et_int
        @ ( image_nat_set_int
          @ ^ [Y2: nat] : ( comple3221217463730067765et_int @ ( image_nat_set_int @ C2 @ ( B3 @ Y2 ) ) )
          @ A3 ) ) ) ).

% UN_UN_flatten
thf(fact_1149_UN__UN__flatten,axiom,
    ! [C2: int > set_nat,B3: nat > set_int,A3: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ C2 @ ( comple3221217463730067765et_int @ ( image_nat_set_int @ B3 @ A3 ) ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [Y2: nat] : ( comple7399068483239264473et_nat @ ( image_int_set_nat @ C2 @ ( B3 @ Y2 ) ) )
          @ A3 ) ) ) ).

% UN_UN_flatten
thf(fact_1150_UN__UN__flatten,axiom,
    ! [C2: int > set_nat,B3: int > set_int,A3: set_int] :
      ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ C2 @ ( comple3221217463730067765et_int @ ( image_int_set_int @ B3 @ A3 ) ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_int_set_nat
          @ ^ [Y2: int] : ( comple7399068483239264473et_nat @ ( image_int_set_nat @ C2 @ ( B3 @ Y2 ) ) )
          @ A3 ) ) ) ).

% UN_UN_flatten
thf(fact_1151_UN__UN__flatten,axiom,
    ! [C2: nat > set_nat,B3: nat > set_nat,A3: set_nat] :
      ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B3 @ A3 ) ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_nat_set_nat
          @ ^ [Y2: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B3 @ Y2 ) ) )
          @ A3 ) ) ) ).

% UN_UN_flatten
thf(fact_1152_UN__UN__flatten,axiom,
    ! [C2: int > set_int,B3: set_int > set_int,A3: set_set_int] :
      ( ( comple3221217463730067765et_int @ ( image_int_set_int @ C2 @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ B3 @ A3 ) ) ) )
      = ( comple3221217463730067765et_int
        @ ( image_524474410958335435et_int
          @ ^ [Y2: set_int] : ( comple3221217463730067765et_int @ ( image_int_set_int @ C2 @ ( B3 @ Y2 ) ) )
          @ A3 ) ) ) ).

% UN_UN_flatten
thf(fact_1153_UN__UN__flatten,axiom,
    ! [C2: nat > set_int,B3: set_int > set_nat,A3: set_set_int] :
      ( ( comple3221217463730067765et_int @ ( image_nat_set_int @ C2 @ ( comple7399068483239264473et_nat @ ( image_4702325430467532143et_nat @ B3 @ A3 ) ) ) )
      = ( comple3221217463730067765et_int
        @ ( image_524474410958335435et_int
          @ ^ [Y2: set_int] : ( comple3221217463730067765et_int @ ( image_nat_set_int @ C2 @ ( B3 @ Y2 ) ) )
          @ A3 ) ) ) ).

% UN_UN_flatten
thf(fact_1154_UN__UN__flatten,axiom,
    ! [C2: set_int > set_int,B3: int > set_set_int,A3: set_int] :
      ( ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ C2 @ ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ B3 @ A3 ) ) ) )
      = ( comple3221217463730067765et_int
        @ ( image_int_set_int
          @ ^ [Y2: int] : ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ C2 @ ( B3 @ Y2 ) ) )
          @ A3 ) ) ) ).

% UN_UN_flatten
thf(fact_1155_UN__UN__flatten,axiom,
    ! [C2: int > set_nat,B3: set_int > set_int,A3: set_set_int] :
      ( ( comple7399068483239264473et_nat @ ( image_int_set_nat @ C2 @ ( comple3221217463730067765et_int @ ( image_524474410958335435et_int @ B3 @ A3 ) ) ) )
      = ( comple7399068483239264473et_nat
        @ ( image_4702325430467532143et_nat
          @ ^ [Y2: set_int] : ( comple7399068483239264473et_nat @ ( image_int_set_nat @ C2 @ ( B3 @ Y2 ) ) )
          @ A3 ) ) ) ).

% UN_UN_flatten
thf(fact_1156_UN__E,axiom,
    ! [B: set_int,B3: set_int > set_set_int,A3: set_set_int] :
      ( ( member_set_int @ B @ ( comple7281953568134767595et_int @ ( image_1010086626112315521et_int @ B3 @ A3 ) ) )
     => ~ ! [X5: set_int] :
            ( ( member_set_int @ X5 @ A3 )
           => ~ ( member_set_int @ B @ ( B3 @ X5 ) ) ) ) ).

% UN_E
thf(fact_1157_UN__E,axiom,
    ! [B: set_int,B3: nat > set_set_int,A3: set_nat] :
      ( ( member_set_int @ B @ ( comple7281953568134767595et_int @ ( image_8927401050382224495et_int @ B3 @ A3 ) ) )
     => ~ ! [X5: nat] :
            ( ( member_nat @ X5 @ A3 )
           => ~ ( member_set_int @ B @ ( B3 @ X5 ) ) ) ) ).

% UN_E
thf(fact_1158_UN__E,axiom,
    ! [B: set_int,B3: int > set_set_int,A3: set_int] :
      ( ( member_set_int @ B @ ( comple7281953568134767595et_int @ ( image_6617583118289273547et_int @ B3 @ A3 ) ) )
     => ~ ! [X5: int] :
            ( ( member_int @ X5 @ A3 )
           => ~ ( member_set_int @ B @ ( B3 @ X5 ) ) ) ) ).

% UN_E
thf(fact_1159_UN__E,axiom,
    ! [B: set_int,B3: ( int > set_int ) > set_set_int,A3: set_int_set_int] :
      ( ( member_set_int @ B @ ( comple7281953568134767595et_int @ ( image_3533601879074506316et_int @ B3 @ A3 ) ) )
     => ~ ! [X5: int > set_int] :
            ( ( member_int_set_int @ X5 @ A3 )
           => ~ ( member_set_int @ B @ ( B3 @ X5 ) ) ) ) ).

% UN_E
thf(fact_1160_UN__lessThan__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_lessThan_UNIV
thf(fact_1161_int_Oadd_Oinj__on__multc,axiom,
    ! [C: int] :
      ( ( member_int @ C @ top_top_set_int )
     => ( inj_on_int_int
        @ ^ [X2: int] : ( plus_plus_int @ X2 @ C )
        @ top_top_set_int ) ) ).

% int.add.inj_on_multc
thf(fact_1162_int_Oadd_Oinj__on__cmult,axiom,
    ! [C: int] :
      ( ( member_int @ C @ top_top_set_int )
     => ( inj_on_int_int @ ( plus_plus_int @ C ) @ top_top_set_int ) ) ).

% int.add.inj_on_cmult
thf(fact_1163_int_Oadd_Oinv__inj,axiom,
    inj_on_int_int @ uminus_uminus_int @ top_top_set_int ).

% int.add.inv_inj
thf(fact_1164_int_Oadd_Oinj__on__g,axiom,
    ! [H2: set_int,A: int] :
      ( ( ord_less_eq_set_int @ H2 @ top_top_set_int )
     => ( ( member_int @ A @ top_top_set_int )
       => ( inj_on_int_int
          @ ^ [Y2: int] : ( plus_plus_int @ Y2 @ A )
          @ H2 ) ) ) ).

% int.add.inj_on_g
thf(fact_1165_i_Oadd_Oderived__set__in__carrier,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 ) ) ) ) )
     => ( ord_less_eq_set_int
        @ ( comple3221217463730067765et_int
          @ ( image_int_set_int
            @ ^ [H1: int] :
                ( comple3221217463730067765et_int
                @ ( image_int_set_int
                  @ ^ [H22: int] : ( insert_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 ) ) ) @ ( 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 ) ) ) @ H1 @ H22 ) @ ( 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 ) ) ) @ H1 ) ) @ ( 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 ) ) ) @ H22 ) ) @ bot_bot_set_int )
                  @ H2 ) )
            @ 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 ) ) ) ) ) ) ).

% i.add.derived_set_in_carrier
thf(fact_1166_i_Oadd_Omono__derived__set,axiom,
    ! [K2: set_int,H2: set_int] :
      ( ( ord_less_eq_set_int @ K2 @ H2 )
     => ( ord_less_eq_set_int
        @ ( comple3221217463730067765et_int
          @ ( image_int_set_int
            @ ^ [H1: int] :
                ( comple3221217463730067765et_int
                @ ( image_int_set_int
                  @ ^ [H22: int] : ( insert_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 ) ) ) @ ( 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 ) ) ) @ H1 @ H22 ) @ ( 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 ) ) ) @ H1 ) ) @ ( 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 ) ) ) @ H22 ) ) @ bot_bot_set_int )
                  @ K2 ) )
            @ K2 ) )
        @ ( comple3221217463730067765et_int
          @ ( image_int_set_int
            @ ^ [H1: int] :
                ( comple3221217463730067765et_int
                @ ( image_int_set_int
                  @ ^ [H22: int] : ( insert_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 ) ) ) @ ( 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 ) ) ) @ H1 @ H22 ) @ ( 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 ) ) ) @ H1 ) ) @ ( 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 ) ) ) @ H22 ) ) @ bot_bot_set_int )
                  @ H2 ) )
            @ H2 ) ) ) ) ).

% i.add.mono_derived_set
thf(fact_1167_i_Oa__rcos__equation,axiom,
    ! [Ha: int,A: int,H: int,B: int,Hb: int] :
      ( ( ( plus_plus_int @ Ha @ A )
        = ( plus_plus_int @ H @ B ) )
     => ( ( member_int @ A @ top_top_set_int )
       => ( ( member_int @ B @ top_top_set_int )
         => ( ( member_int @ H @ i )
           => ( ( member_int @ Ha @ i )
             => ( ( member_int @ Hb @ i )
               => ( member_int @ ( plus_plus_int @ Hb @ A )
                  @ ( comple3221217463730067765et_int
                    @ ( image_int_set_int
                      @ ^ [H3: int] : ( insert_int @ ( plus_plus_int @ H3 @ B ) @ bot_bot_set_int )
                      @ i ) ) ) ) ) ) ) ) ) ).

% i.a_rcos_equation
thf(fact_1168_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_1169_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_1170_Sup__bool__def,axiom,
    ( complete_Sup_Sup_o
    = ( member_o @ $true ) ) ).

% Sup_bool_def
thf(fact_1171_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat
thf(fact_1172_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_1173_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_1174_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_1175_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_1176_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_1177_int_Osubset__Idl__subset,axiom,
    ! [I5: set_int,H2: set_int] :
      ( ( ord_less_eq_set_int @ I5 @ top_top_set_int )
     => ( ( ord_less_eq_set_int @ H2 @ I5 )
       => ( 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 ) ) ) @ I5 ) ) ) ) ).

% int.subset_Idl_subset
thf(fact_1178_int_Ogenideal__self,axiom,
    ! [S3: set_int] :
      ( ( ord_less_eq_set_int @ S3 @ top_top_set_int )
     => ( ord_less_eq_set_int @ S3 @ ( 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 ) ) ) @ S3 ) ) ) ).

% int.genideal_self
thf(fact_1179_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_1180_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_1181_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_1182_int__Idl__subset__ideal,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_set_int @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ K @ bot_bot_set_int ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ L @ bot_bot_set_int ) ) )
      = ( member_int @ K @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ L @ bot_bot_set_int ) ) ) ) ).

% int_Idl_subset_ideal
thf(fact_1183_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_1184_int__cosetI,axiom,
    ! [N2: nat,U: int,V: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( modulo_modulo_int @ U @ ( semiri1314217659103216013at_int @ N2 ) )
          = ( modulo_modulo_int @ V @ ( semiri1314217659103216013at_int @ N2 ) ) )
       => ( ( 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 @ N2 ) @ bot_bot_set_int ) ) @ U )
          = ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ ( semiri1314217659103216013at_int @ N2 ) @ bot_bot_set_int ) ) @ V ) ) ) ) ).

% int_cosetI
thf(fact_1185__092_060open_062_II_A_L_062_092_060_094bsub_062_092_060Z_062_092_060_094esub_062_Aint_Ax_H_J_A_092_060oplus_062_092_060_094bsub_062ZFact_A_Iint_An_J_092_060_094esub_062_A_II_A_L_062_092_060_094bsub_062_092_060Z_062_092_060_094esub_062_Aint_Ay_H_J_A_061_AI_A_L_062_092_060_094bsub_062_092_060Z_062_092_060_094esub_062_Aint_Ax_H_A_L_Aint_Ay_H_092_060close_062,axiom,
    ( ( 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 @ 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 @ ( semiri1314217659103216013at_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 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ x ) @ ( semiri1314217659103216013at_int @ y ) ) ) ) ).

% \<open>(I +>\<^bsub>\<Z>\<^esub> int x') \<oplus>\<^bsub>ZFact (int n)\<^esub> (I +>\<^bsub>\<Z>\<^esub> int y') = I +>\<^bsub>\<Z>\<^esub> int x' + int y'\<close>
thf(fact_1186_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_1187_r_Oadd__additive__subgroups,axiom,
    ! [H2: set_set_int,K2: set_set_int] :
      ( ( additi7073586575563672860t_unit @ H2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
     => ( ( additi7073586575563672860t_unit @ K2 @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
       => ( additi7073586575563672860t_unit @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 @ K2 ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).

% r.add_additive_subgroups
thf(fact_1188_x__carr,axiom,
    member_set_int @ x2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).

% x_carr
thf(fact_1189_y__carr,axiom,
    member_set_int @ y2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ).

% y_carr
thf(fact_1190_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_1191_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_1192_r_Oa__rcos__assoc__lcos,axiom,
    ! [H2: set_set_int,K2: set_set_int,X: set_int] :
      ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
     => ( ( ord_le4403425263959731960et_int @ K2 @ ( 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 ) @ K2 )
            = ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ K2 ) ) ) ) ) ) ).

% r.a_rcos_assoc_lcos
thf(fact_1193_r_Oa__setmult__rcos__assoc,axiom,
    ! [H2: set_set_int,K2: set_set_int,X: set_int] :
      ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
     => ( ( ord_le4403425263959731960et_int @ K2 @ ( 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 ) ) @ K2 @ X ) )
            = ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 @ K2 ) @ X ) ) ) ) ) ).

% r.a_setmult_rcos_assoc
thf(fact_1194_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_1195_r_Oa__lcos__mult__one,axiom,
    ! [M3: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ M3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
     => ( ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ M3 )
        = M3 ) ) ).

% r.a_lcos_mult_one
thf(fact_1196_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_1197_r_Oset__add__closed,axiom,
    ! [A3: set_set_int,B3: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ A3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
     => ( ( ord_le4403425263959731960et_int @ B3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
       => ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A3 @ B3 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).

% r.set_add_closed
thf(fact_1198_r_Oset__add__comm,axiom,
    ! [I5: set_set_int,J4: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ I5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
     => ( ( ord_le4403425263959731960et_int @ J4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
       => ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ I5 @ J4 )
          = ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ J4 @ I5 ) ) ) ) ).

% r.set_add_comm
thf(fact_1199_r_Osetadd__subset__G,axiom,
    ! [H2: set_set_int,K2: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
     => ( ( ord_le4403425263959731960et_int @ K2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
       => ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H2 @ K2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).

% r.setadd_subset_G
thf(fact_1200_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_1201_r_Ocarrier__not__empty,axiom,
    ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) )
   != bot_bot_set_set_int ) ).

% r.carrier_not_empty
thf(fact_1202_r_Oadd_Omono__derived__set,axiom,
    ! [K2: set_set_int,H2: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ K2 @ H2 )
     => ( ord_le4403425263959731960et_int
        @ ( comple7281953568134767595et_int
          @ ( image_1010086626112315521et_int
            @ ^ [H1: set_int] :
                ( comple7281953568134767595et_int
                @ ( image_1010086626112315521et_int
                  @ ^ [H22: set_int] : ( insert_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H1 @ H22 ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H1 ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H22 ) ) @ bot_bot_set_set_int )
                  @ K2 ) )
            @ K2 ) )
        @ ( comple7281953568134767595et_int
          @ ( image_1010086626112315521et_int
            @ ^ [H1: set_int] :
                ( comple7281953568134767595et_int
                @ ( image_1010086626112315521et_int
                  @ ^ [H22: set_int] : ( insert_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H1 @ H22 ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H1 ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H22 ) ) @ bot_bot_set_set_int )
                  @ H2 ) )
            @ H2 ) ) ) ) ).

% r.add.mono_derived_set
thf(fact_1203_r_Oa__coset__add__inv1,axiom,
    ! [M3: set_set_int,X: set_int,Y: set_int] :
      ( ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M3 @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) )
        = M3 )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
       => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
         => ( ( ord_le4403425263959731960et_int @ M3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
           => ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M3 @ X )
              = ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M3 @ Y ) ) ) ) ) ) ).

% r.a_coset_add_inv1
thf(fact_1204_r_Oa__coset__add__inv2,axiom,
    ! [M3: set_set_int,X: set_int,Y: set_int] :
      ( ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M3 @ X )
        = ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M3 @ Y ) )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
       => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
         => ( ( ord_le4403425263959731960et_int @ M3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
           => ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M3 @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y ) ) )
              = M3 ) ) ) ) ) ).

% r.a_coset_add_inv2
thf(fact_1205_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 )
       => ( ! [X5: set_int] :
              ( ( member_set_int @ X5 @ H2 )
             => ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X5 ) @ H2 ) )
         => ( ! [X5: set_int] :
                ( ( member_set_int @ X5 @ H2 )
               => ! [Xa: set_int] :
                    ( ( member_set_int @ Xa @ H2 )
                   => ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X5 @ Xa ) @ H2 ) ) )
           => ( member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ H2 ) ) ) ) ) ).

% r.add.one_in_subset
thf(fact_1206_r_Oa__coset__add__assoc,axiom,
    ! [M3: set_set_int,G: set_int,H: set_int] :
      ( ( ord_le4403425263959731960et_int @ M3 @ ( 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 ) ) @ M3 @ G ) @ H )
            = ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M3 @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ G @ H ) ) ) ) ) ) ).

% r.a_coset_add_assoc
thf(fact_1207_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_1208_r_Oa__lcos__m__assoc,axiom,
    ! [M3: set_set_int,G: set_int,H: set_int] :
      ( ( ord_le4403425263959731960et_int @ M3 @ ( 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 @ M3 ) )
            = ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ G @ H ) @ M3 ) ) ) ) ) ).

% r.a_lcos_m_assoc
thf(fact_1209_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_1210_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_1211_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_1212_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_1213_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_1214_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_1215_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_1216_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_1217_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_1218_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_1219_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_1220_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_1221_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_1222_r_Ominus__unique,axiom,
    ! [Y: set_int,X: set_int,Y3: 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 @ Y3 )
          = ( 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 @ Y3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
             => ( Y = Y3 ) ) ) ) ) ) ).

% r.minus_unique
thf(fact_1223_r_Oadd_Or__inv__ex,axiom,
    ! [X: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
     => ? [X5: set_int] :
          ( ( member_set_int @ X5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
          & ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X @ X5 )
            = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).

% r.add.r_inv_ex
thf(fact_1224_r_Oadd_Oone__unique,axiom,
    ! [U: set_int] :
      ( ( member_set_int @ U @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
     => ( ! [X5: set_int] :
            ( ( member_set_int @ X5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
           => ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ U @ X5 )
              = X5 ) )
       => ( U
          = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).

% r.add.one_unique
thf(fact_1225_r_Oadd_Ol__inv__ex,axiom,
    ! [X: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
     => ? [X5: set_int] :
          ( ( member_set_int @ X5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
          & ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X5 @ X )
            = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).

% r.add.l_inv_ex
thf(fact_1226_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_1227_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_1228_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_1229_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_1230_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_1231_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_1232_r_Oadd_Oderived__set__in__carrier,axiom,
    ! [H2: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
     => ( ord_le4403425263959731960et_int
        @ ( comple7281953568134767595et_int
          @ ( image_1010086626112315521et_int
            @ ^ [H1: set_int] :
                ( comple7281953568134767595et_int
                @ ( image_1010086626112315521et_int
                  @ ^ [H22: set_int] : ( insert_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H1 @ H22 ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H1 ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ H22 ) ) @ bot_bot_set_set_int )
                  @ H2 ) )
            @ H2 ) )
        @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).

% r.add.derived_set_in_carrier
thf(fact_1233_r_Oadd_Oinj__on__multc,axiom,
    ! [C: set_int] :
      ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
     => ( inj_on6435365835345961783et_int
        @ ^ [X2: set_int] : ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ X2 @ C )
        @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).

% r.add.inj_on_multc
thf(fact_1234_r_Oadd_Oinj__on__cmult,axiom,
    ! [C: set_int] :
      ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
     => ( inj_on6435365835345961783et_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ C ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).

% r.add.inj_on_cmult
thf(fact_1235_r_Oadd_Osurj__const__mult,axiom,
    ! [A: set_int] :
      ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
     => ( ( image_524474410958335435et_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ A ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
        = ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ).

% r.add.surj_const_mult
thf(fact_1236_r_Oadd_Oinj__on__g,axiom,
    ! [H2: set_set_int,A: set_int] :
      ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
     => ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
       => ( inj_on6435365835345961783et_int
          @ ^ [Y2: set_int] : ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ Y2 @ A )
          @ H2 ) ) ) ).

% r.add.inj_on_g
thf(fact_1237_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_1238_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_1239_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_1240_r_Oa__coset__add__zero,axiom,
    ! [M3: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ M3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
     => ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ M3 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) )
        = M3 ) ) ).

% r.a_coset_add_zero
thf(fact_1241_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_1242_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_1243_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_1244_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_1245_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_1246_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_1247_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_1248_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_1249_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_1250_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_1251_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_1252_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_1253_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_1254_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_1255_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_1256__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_1257_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_1258_fin__zfact,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% fin_zfact
thf(fact_1259_ZFact__zero,axiom,
    ( ( partia966996272515721803t_unit @ ( zFact @ zero_zero_int ) )
    = ( comple7281953568134767595et_int
      @ ( image_6617583118289273547et_int
        @ ^ [A4: int] : ( insert_set_int @ ( insert_int @ A4 @ bot_bot_set_int ) @ bot_bot_set_set_int )
        @ top_top_set_int ) ) ) ).

% ZFact_zero
thf(fact_1260_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_1261_r_Oonepideal,axiom,
    princi8860937869964495385t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ).

% r.onepideal
thf(fact_1262_x_H__def,axiom,
    ( x
    = ( zfact_iso_inv @ n @ x2 ) ) ).

% x'_def
thf(fact_1263_y_H__def,axiom,
    ( y
    = ( zfact_iso_inv @ n @ y2 ) ) ).

% y'_def
thf(fact_1264_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_1265__092_060open_062x_A_092_060oplus_062_092_060_094bsub_062ZFact_A_Iint_An_J_092_060_094esub_062_Ay_A_061_A_II_A_L_062_092_060_094bsub_062_092_060Z_062_092_060_094esub_062_Aint_Ax_H_J_A_092_060oplus_062_092_060_094bsub_062ZFact_A_Iint_An_J_092_060_094esub_062_A_II_A_L_062_092_060_094bsub_062_092_060Z_062_092_060_094esub_062_Aint_Ay_H_J_092_060close_062,axiom,
    ( ( 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 @ 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 @ ( semiri1314217659103216013at_int @ y ) ) ) ) ).

% \<open>x \<oplus>\<^bsub>ZFact (int n)\<^esub> y = (I +>\<^bsub>\<Z>\<^esub> int x') \<oplus>\<^bsub>ZFact (int n)\<^esub> (I +>\<^bsub>\<Z>\<^esub> int y')\<close>
thf(fact_1266_calculation,axiom,
    ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) @ x2 @ y2 )
    = ( 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 @ ( modulo_modulo_nat @ ( plus_plus_nat @ x @ y ) @ n ) ) ) ) ).

% calculation
thf(fact_1267_zfact__iso__inv__0,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( zfact_iso_inv @ N2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N2 ) ) ) )
        = zero_zero_nat ) ) ).

% zfact_iso_inv_0
thf(fact_1268_zfact__coset,axiom,
    ! [N2: nat,X: set_int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ N2 ) ) ) )
       => ( 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 @ N2 ) @ bot_bot_set_int ) ) @ ( semiri1314217659103216013at_int @ ( zfact_iso_inv @ N2 @ X ) ) ) ) ) ) ).

% zfact_coset
thf(fact_1269_r_OboundD__carrier,axiom,
    ! [N2: nat,F: nat > set_int,M: nat] :
      ( ( bound_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) @ N2 @ F )
     => ( ( ord_less_nat @ N2 @ M )
       => ( member_set_int @ ( F @ M ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ n ) ) ) ) ) ) ).

% r.boundD_carrier
thf(fact_1270_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_1271_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_1272_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

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( 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 @ ( modulo_modulo_nat @ ( plus_plus_nat @ x @ y ) @ 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
        @ ( add_nat_Product_unit
          @ ( partia8414553997885618600t_unit @ ( set_ord_lessThan_nat @ n )
            @ ( monoid7204241362306134736t_unit
              @ ^ [X2: nat,Y2: nat] : ( modulo_modulo_nat @ ( times_times_nat @ X2 @ Y2 ) @ n )
              @ one_one_nat
              @ ( ring_e8156451031480216102t_unit @ zero_zero_nat
                @ ^ [X2: nat,Y2: nat] : ( modulo_modulo_nat @ ( plus_plus_nat @ X2 @ Y2 ) @ n )
                @ product_Unity ) ) )
          @ x
          @ y ) ) ) ) ).

%------------------------------------------------------------------------------