%------------------------------------------------------------------------------
% 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_00116_003959__18264136_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1279 ( 614 unt;  63 typ;   0 def)
%            Number of atoms       : 3089 (1293 equ;   0 cnn)
%            Maximal formula atoms :    8 (   2 avg)
%            Number of connectives : 12708 ( 215   ~;  66   |; 125   &;11165   @)
%                                         (   0 <=>;1137  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   20 (   6 avg)
%            Number of types       :   10 (   9 usr)
%            Number of type conns  :  165 ( 165   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   57 (  54 usr;  13 con; 0-3 aty)
%            Number of variables   : 2715 (  68   ^;2594   !;  53   ?;2715   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:41:01.736
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
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__Int__Oint_Mt__Ring__Oring__Oring____ext_It__Int__Oint_Mt__Product____Type__Ounit_J_J,type,
    monoid8431999971278595628t_unit: $tType ).

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

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
    set_int: $tType ).

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

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

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

% Explicit typings (54)
thf(sy_c_AbelCoset_Oa__r__coset_001t__Int__Oint_001t__Product____Type__Ounit,type,
    a_r_co6205493800230438172t_unit: partia2818514838349642498t_unit > set_int > int > set_int ).

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

thf(sy_c_Congruence_Opartial__object_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_Field_Omod__ring,type,
    mod_ring: nat > partia4692342223508353374t_unit ).

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__Nat__Onat_001t__Nat__Onat,type,
    inj_on_nat_nat: ( nat > nat ) > set_nat > $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_Omult_001t__Nat__Onat_001t__Ring__Oring__Oring____ext_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
    mult_n6028127365542633569t_unit: partia4692342223508353374t_unit > nat > nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Int__Oint_M_Eo_J,type,
    minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
    minus_minus_int: int > int > int ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
    minus_minus_set_int: set_int > set_int > set_int ).

thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
    one_one_int: int ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

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

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

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

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

thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Int__Oint_M_Eo_J,type,
    uminus_uminus_int_o: ( int > $o ) > int > $o ).

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_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_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_IntRing_OZMod,type,
    zMod: int > int > set_int ).

thf(sy_c_Nat_OSuc,type,
    suc: nat > nat ).

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

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

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_Eo_J,type,
    bot_bot_int_o: int > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

thf(sy_c_Orderings_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__eq_001_062_It__Int__Oint_M_Eo_J,type,
    ord_less_eq_int_o: ( int > $o ) > ( int > $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_It__Int__Oint_J,type,
    ord_less_eq_set_int: set_int > set_int > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Int__Oint_M_Eo_J,type,
    top_top_int_o: int > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

thf(sy_c_Product__Type_OUnity,type,
    product_Unity: product_unit ).

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_Oadd__pow_001t__Int__Oint_001t__Product____Type__Ounit_001t__Int__Oint,type,
    add_po6254319245437977817it_int: partia2818514838349642498t_unit > int > int > int ).

thf(sy_c_Ring_Oadd__pow_001t__Int__Oint_001t__Product____Type__Ounit_001t__Nat__Onat,type,
    add_po6256809715947028093it_nat: partia2818514838349642498t_unit > nat > int > int ).

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_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_Ozero_001t__Int__Oint_001t__Product____Type__Ounit,type,
    zero_i2266321264637750939t_unit: partia2818514838349642498t_unit > int ).

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

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

thf(sy_c_member_001t__Int__Oint,type,
    member_int: int > set_int > $o ).

thf(sy_v_n,type,
    n: nat ).

thf(sy_v_x_H____,type,
    x: nat ).

thf(sy_v_y_H____,type,
    y: nat ).

% Relevant facts (1212)
thf(fact_0_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_1_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_2_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_3_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_4_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_5_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_6_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_7_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_8_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_9_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_10_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_11_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_12_UNIV_I3_J,axiom,
    ! [P: int > $o] :
      ( ( ! [X2: int] :
            ( ( member_int @ X2 @ top_top_set_int )
           => ( P @ X2 ) ) )
      = ( ! [X3: int] : ( P @ X3 ) ) ) ).

% UNIV(3)
thf(fact_13_UNIV_I4_J,axiom,
    ! [P: int > $o] :
      ( ( ? [X2: int] :
            ( ( member_int @ X2 @ top_top_set_int )
            & ( P @ X2 ) ) )
      = ( ? [X3: int] : ( P @ X3 ) ) ) ).

% UNIV(4)
thf(fact_14_int_Oadd_Oone__closed,axiom,
    member_int @ zero_zero_int @ top_top_set_int ).

% int.add.one_closed
thf(fact_15_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_16_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_17_int_Oone__closed,axiom,
    member_int @ one_one_int @ top_top_set_int ).

% int.one_closed
thf(fact_18_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_19_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_20_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_21_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_22_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_23_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_24_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_25_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_26_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_27_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_28_int_Ozero__not__one,axiom,
    zero_zero_int != one_one_int ).

% int.zero_not_one
thf(fact_29_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_30_int_Oadd_Ocarrier__not__empty,axiom,
    top_top_set_int != bot_bot_set_int ).

% int.add.carrier_not_empty
thf(fact_31_int_Oadd_Oone__unique,axiom,
    ! [U: int] :
      ( ( member_int @ U @ top_top_set_int )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ top_top_set_int )
           => ( ( plus_plus_int @ U @ X4 )
              = X4 ) )
       => ( U = zero_zero_int ) ) ) ).

% int.add.one_unique
thf(fact_32_int_Oadd_Oinv__unique,axiom,
    ! [Y: int,X: int,Y2: int] :
      ( ( ( plus_plus_int @ Y @ X )
        = zero_zero_int )
     => ( ( ( plus_plus_int @ X @ Y2 )
          = zero_zero_int )
       => ( ( member_int @ X @ top_top_set_int )
         => ( ( member_int @ Y @ top_top_set_int )
           => ( ( member_int @ Y2 @ top_top_set_int )
             => ( Y = Y2 ) ) ) ) ) ) ).

% int.add.inv_unique
thf(fact_33_int_Oadd_Or__inv__ex,axiom,
    ! [X: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ? [X4: int] :
          ( ( member_int @ X4 @ top_top_set_int )
          & ( ( plus_plus_int @ X @ X4 )
            = zero_zero_int ) ) ) ).

% int.add.r_inv_ex
thf(fact_34_int_Oadd_Ol__inv__ex,axiom,
    ! [X: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ? [X4: int] :
          ( ( member_int @ X4 @ top_top_set_int )
          & ( ( plus_plus_int @ X4 @ X )
            = zero_zero_int ) ) ) ).

% int.add.l_inv_ex
thf(fact_35_int_Oadd_Oinv__comm,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ X @ Y )
        = zero_zero_int )
     => ( ( member_int @ X @ top_top_set_int )
       => ( ( member_int @ Y @ top_top_set_int )
         => ( ( plus_plus_int @ Y @ X )
            = zero_zero_int ) ) ) ) ).

% int.add.inv_comm
thf(fact_36_int_Ointegral__iff,axiom,
    ! [A: int,B: int] :
      ( ( member_int @ A @ top_top_set_int )
     => ( ( member_int @ B @ top_top_set_int )
       => ( ( ( times_times_int @ A @ B )
            = zero_zero_int )
          = ( ( A = zero_zero_int )
            | ( B = zero_zero_int ) ) ) ) ) ).

% int.integral_iff
thf(fact_37_int_Om__rcancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( member_int @ A @ top_top_set_int )
       => ( ( member_int @ B @ top_top_set_int )
         => ( ( member_int @ C @ top_top_set_int )
           => ( ( ( times_times_int @ B @ A )
                = ( times_times_int @ C @ A ) )
              = ( B = C ) ) ) ) ) ) ).

% int.m_rcancel
thf(fact_38_int_Om__lcancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( member_int @ A @ top_top_set_int )
       => ( ( member_int @ B @ top_top_set_int )
         => ( ( member_int @ C @ top_top_set_int )
           => ( ( ( times_times_int @ A @ B )
                = ( times_times_int @ A @ C ) )
              = ( B = C ) ) ) ) ) ) ).

% int.m_lcancel
thf(fact_39_int_Ointegral,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
     => ( ( member_int @ A @ top_top_set_int )
       => ( ( member_int @ B @ top_top_set_int )
         => ( ( A = zero_zero_int )
            | ( B = zero_zero_int ) ) ) ) ) ).

% int.integral
thf(fact_40_int_Or__distr,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( member_int @ Y @ top_top_set_int )
       => ( ( member_int @ Z @ top_top_set_int )
         => ( ( times_times_int @ Z @ ( plus_plus_int @ X @ Y ) )
            = ( plus_plus_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) ) ) ) ) ) ).

% int.r_distr
thf(fact_41_int_Ol__distr,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( member_int @ Y @ top_top_set_int )
       => ( ( member_int @ Z @ top_top_set_int )
         => ( ( times_times_int @ ( plus_plus_int @ X @ Y ) @ Z )
            = ( plus_plus_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) ) ) ) ) ) ).

% int.l_distr
thf(fact_42_int_Oone__unique,axiom,
    ! [U: int] :
      ( ( member_int @ U @ top_top_set_int )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ top_top_set_int )
           => ( ( times_times_int @ U @ X4 )
              = X4 ) )
       => ( U = one_one_int ) ) ) ).

% int.one_unique
thf(fact_43_int_Oinv__unique,axiom,
    ! [Y: int,X: int,Y2: int] :
      ( ( ( times_times_int @ Y @ X )
        = one_one_int )
     => ( ( ( times_times_int @ X @ Y2 )
          = one_one_int )
       => ( ( member_int @ X @ top_top_set_int )
         => ( ( member_int @ Y @ top_top_set_int )
           => ( ( member_int @ Y2 @ top_top_set_int )
             => ( Y = Y2 ) ) ) ) ) ) ).

% int.inv_unique
thf(fact_44_int_Ocarrier__one__not__zero,axiom,
    ( ( top_top_set_int
     != ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
    = ( one_one_int != zero_zero_int ) ) ).

% int.carrier_one_not_zero
thf(fact_45_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
    ! [A2: set_int] :
      ( ( collect_int
        @ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_48_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_49_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_50_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_51_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_52_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_53_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_54_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_55_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_56_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( ( times_times_int @ A @ C )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_57_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_58_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( ( times_times_int @ C @ A )
        = C )
      = ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_59_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_60_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_61_ZMod__def,axiom,
    ( zMod
    = ( ^ [K2: int] : ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ K2 @ bot_bot_set_int ) ) ) ) ) ).

% ZMod_def
thf(fact_62_i_Oadd_Oint__pow__pow,axiom,
    ! [X: int,M: int,N: 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_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) )
        = ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( times_times_int @ N @ M ) @ X ) ) ) ).

% i.add.int_pow_pow
thf(fact_63_n__ge__1,axiom,
    ord_less_nat @ one_one_nat @ n ).

% n_ge_1
thf(fact_64_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_65_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_66_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_67_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_68_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_69_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_70_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_71_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_72_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_73_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_74_i_Oadd_Oint__pow__distrib,axiom,
    ! [X: int,Y: int,I: 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_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ ( 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 ) ) ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ Y ) ) ) ) ) ).

% i.add.int_pow_distrib
thf(fact_75_i_Oadd_Oint__pow__mult,axiom,
    ! [X: int,I: int,J: 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_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( plus_plus_int @ I @ J ) @ 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 ) ) ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ J @ X ) ) ) ) ).

% i.add.int_pow_mult
thf(fact_76_i_Oadd_Oint__pow__mult__distrib,axiom,
    ! [X: int,Y: int,I: 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 )
        = ( 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 ) )
     => ( ( 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_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ ( 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 ) ) ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ Y ) ) ) ) ) ) ).

% i.add.int_pow_mult_distrib
thf(fact_77_int_Oadd_Oint__pow__1,axiom,
    ! [X: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ one_one_int @ X )
        = X ) ) ).

% int.add.int_pow_1
thf(fact_78_int_Oadd_Oint__pow__one,axiom,
    ! [Z: int] :
      ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Z @ zero_zero_int )
      = zero_zero_int ) ).

% int.add.int_pow_one
thf(fact_79_int_Oadd_Oint__pow__closed,axiom,
    ! [X: int,I: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( member_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X ) @ top_top_set_int ) ) ).

% int.add.int_pow_closed
thf(fact_80_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_81_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_82_i_Oadd_Oint__pow__1,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_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ one_one_int @ X )
        = X ) ) ).

% i.add.int_pow_1
thf(fact_83_i_Oadd_Oint__pow__closed,axiom,
    ! [X: int,I: 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 @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ 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.int_pow_closed
thf(fact_84_rcos__zfact,axiom,
    ! [K: int,L: int,R: int] :
      ( ( member_int @ K @ ( zMod @ L @ R ) )
     => ? [X4: int] :
          ( K
          = ( plus_plus_int @ ( times_times_int @ X4 @ L ) @ R ) ) ) ).

% rcos_zfact
thf(fact_85_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_86_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_87_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_88_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_89_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_90_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_91_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_92_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_93_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_94_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_95_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_96_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_97_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_98_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_99_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_100_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_101_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_102_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_103_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_104_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_105_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_106_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_107_int_Oadd_Oint__pow__pow,axiom,
    ! [X: int,M: int,N: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) )
        = ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( times_times_int @ N @ M ) @ X ) ) ) ).

% int.add.int_pow_pow
thf(fact_108_int_Oadd_Oint__pow__mult,axiom,
    ! [X: int,I: int,J: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( plus_plus_int @ I @ J ) @ X )
        = ( plus_plus_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ J @ X ) ) ) ) ).

% int.add.int_pow_mult
thf(fact_109_int_Oadd__pow__ldistr__int,axiom,
    ! [A: int,B: int,K: int] :
      ( ( member_int @ A @ top_top_set_int )
     => ( ( member_int @ B @ top_top_set_int )
       => ( ( times_times_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K @ A ) @ B )
          = ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K @ ( times_times_int @ A @ B ) ) ) ) ) ).

% int.add_pow_ldistr_int
thf(fact_110_int_Oadd__pow__rdistr__int,axiom,
    ! [A: int,B: int,K: int] :
      ( ( member_int @ A @ top_top_set_int )
     => ( ( member_int @ B @ top_top_set_int )
       => ( ( times_times_int @ A @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K @ B ) )
          = ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K @ ( times_times_int @ A @ B ) ) ) ) ) ).

% int.add_pow_rdistr_int
thf(fact_111_int_Oadd_Oint__pow__distrib,axiom,
    ! [X: int,Y: int,I: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( member_int @ Y @ top_top_set_int )
       => ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ ( plus_plus_int @ X @ Y ) )
          = ( plus_plus_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ Y ) ) ) ) ) ).

% int.add.int_pow_distrib
thf(fact_112_int_Oadd_Oint__pow__mult__distrib,axiom,
    ! [X: int,Y: int,I: int] :
      ( ( ( plus_plus_int @ X @ Y )
        = ( plus_plus_int @ Y @ X ) )
     => ( ( member_int @ X @ top_top_set_int )
       => ( ( member_int @ Y @ top_top_set_int )
         => ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ ( plus_plus_int @ X @ Y ) )
            = ( plus_plus_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ Y ) ) ) ) ) ) ).

% int.add.int_pow_mult_distrib
thf(fact_113_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_114_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_115_i_Oadd_Oinv__unique,axiom,
    ! [Y: int,X: int,Y2: int] :
      ( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Y @ X )
        = ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
     => ( ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ Y2 )
          = ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
       => ( ( member_int @ X @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
         => ( ( member_int @ Y @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
           => ( ( member_int @ Y2 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
             => ( Y = Y2 ) ) ) ) ) ) ).

% i.add.inv_unique
thf(fact_116_i_Oadd_Ol__inv__ex,axiom,
    ! [X: int] :
      ( ( member_int @ X @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
     => ? [X4: int] :
          ( ( member_int @ X4 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
          & ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X4 @ X )
            = ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ) ).

% i.add.l_inv_ex
thf(fact_117_i_Oadd_Oone__unique,axiom,
    ! [U: int] :
      ( ( member_int @ U @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
           => ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ U @ X4 )
              = X4 ) )
       => ( U
          = ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ) ).

% i.add.one_unique
thf(fact_118_i_Oadd_Or__inv__ex,axiom,
    ! [X: int] :
      ( ( member_int @ X @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
     => ? [X4: int] :
          ( ( member_int @ X4 @ ( partia8426541738980984321t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) )
          & ( ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X @ X4 )
            = ( zero_i2266321264637750939t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) ) ) ) ) ).

% i.add.r_inv_ex
thf(fact_119_i_Oadd_Oint__pow__inv,axiom,
    ! [X: int,I: 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_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ ( 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 ) ) ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X ) ) ) ) ).

% i.add.int_pow_inv
thf(fact_120_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_121_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_122_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_123_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_124_n__ge__0,axiom,
    ord_less_nat @ zero_zero_nat @ n ).

% n_ge_0
thf(fact_125_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_126_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_127_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_128_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_129_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_130_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_131_i_Oadd_Oint__pow__one,axiom,
    ! [Z: int] :
      ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ Z @ ( 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_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.int_pow_one
thf(fact_132_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_133_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_134_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_135_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_136_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_137_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_138_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_139_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_140_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_141_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_142_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_143_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_144_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_145_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_146_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_147_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_148_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_149_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_150_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_151_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_152_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_153_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_154_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_155_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_156_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_157_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_158_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_159_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_160_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_161_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_162_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_163_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_164_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_165_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_166_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_167_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_168_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_169_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_170_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_171_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_172_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_173_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_174_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_175_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_176_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_177_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_178_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_179_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_180_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_181_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_182_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_183_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_184_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_185_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_186_i_Oadd_Oone__in__subset,axiom,
    ! [H: set_int] :
      ( ( ord_less_eq_set_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 ) ) ) ) )
     => ( ( H != bot_bot_set_int )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ H )
             => ( member_int @ ( a_inv_8811962894454695315t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X4 ) @ H ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ H )
               => ! [Xa: int] :
                    ( ( member_int @ Xa @ H )
                   => ( member_int @ ( add_int_Product_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ X4 @ Xa ) @ H ) ) )
           => ( 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 ) ) ) ) @ H ) ) ) ) ) ).

% i.add.one_in_subset
thf(fact_187_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_188_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_189_i_Oadd_Oint__pow__diff,axiom,
    ! [X: int,N: int,M: 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_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( minus_minus_int @ N @ M ) @ 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 ) ) ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ 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 ) ) ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X ) ) ) ) ) ).

% i.add.int_pow_diff
thf(fact_190_i_Oadd_Oint__pow__neg,axiom,
    ! [X: int,I: 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_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( uminus_uminus_int @ I ) @ 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 ) ) ) @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X ) ) ) ) ).

% i.add.int_pow_neg
thf(fact_191_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N )
        = one_one_nat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_192_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1314217659103216013at_int @ N )
        = one_one_int )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_193_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_194_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_195_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_196_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_197_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_198_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_199_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_200_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_201_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_202_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_203_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_204_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_205_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_206_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_207_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_208_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_209_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_210_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_211_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_212_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_213_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N @ K ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_214_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_215_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N ) )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_216_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_217_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_218_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_219_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_220_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_221_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( ( plus_plus_int @ B @ A )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_222_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_223_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_224_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_225_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ B @ A ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_226_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_227_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ A @ B ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_228_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_229_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_230_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_231_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_232_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_233_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_234_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_235_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_236_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_237_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_238_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_239_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_add
thf(fact_240_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_add
thf(fact_241_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_242_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_243_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_244_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_245_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mult
thf(fact_246_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mult
thf(fact_247_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_248_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_249_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_250_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_251_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_252_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_253_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_254_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_255_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_256_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_257_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_258_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_259_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_260_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_261_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_262_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_263_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_264_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_265_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_266_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_267_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_268_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_269_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_270_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( uminus_uminus_int @ A ) )
      = ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_271_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
thf(fact_272_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_273_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_274_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_275_add__minus__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_276_minus__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_277_minus__add__distrib,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).

% minus_add_distrib
thf(fact_278_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_279_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_280_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_281_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_282_minus__diff__eq,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
      = ( minus_minus_int @ B @ A ) ) ).

% minus_diff_eq
thf(fact_283_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_284_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_285_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_286_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% mult_less_cancel2
thf(fact_287_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_288_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% nat_0_less_mult_iff
thf(fact_289_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_290_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_291_int_Ominus__zero,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% int.minus_zero
thf(fact_292_int_Ominus__closed,axiom,
    ! [X: int,Y: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( member_int @ Y @ top_top_set_int )
       => ( member_int @ ( minus_minus_int @ X @ Y ) @ top_top_set_int ) ) ) ).

% int.minus_closed
thf(fact_293_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_294_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_295_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_296_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_297_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel1
thf(fact_298_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_299_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel2
thf(fact_300_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_301_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_302_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_303_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_304_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_305_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_306_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_307_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_308_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel1
thf(fact_309_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_310_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel2
thf(fact_311_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_312_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_313_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_314_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_315_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_316_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_317_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_318_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% neg_0_le_iff_le
thf(fact_319_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_le_0_iff_le
thf(fact_320_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% less_eq_neg_nonpos
thf(fact_321_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_322_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_323_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_324_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_325_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_326_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_327_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_328_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_329_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_330_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_331_less__neg__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% less_neg_neg
thf(fact_332_neg__less__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_pos
thf(fact_333_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% neg_0_less_iff_less
thf(fact_334_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% neg_less_0_iff_less
thf(fact_335_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_336_diff__minus__eq__add,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
      = ( plus_plus_int @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_337_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
      = ( minus_minus_int @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_338_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_339_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_340_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_341_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_342_int_Oa__coset__add__zero,axiom,
    ! [M2: set_int] :
      ( ( ord_less_eq_set_int @ M2 @ 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 ) ) ) @ M2 @ zero_zero_int )
        = M2 ) ) ).

% int.a_coset_add_zero
thf(fact_343_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% less_minus_iff
thf(fact_344_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_345_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D @ C )
       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_346_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_347_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_348_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_349_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_350_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_351_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
         => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_352_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_353_int_Ominus__eq,axiom,
    ( minus_minus_int
    = ( ^ [X2: int,Y3: int] : ( plus_plus_int @ X2 @ ( uminus_uminus_int @ Y3 ) ) ) ) ).

% int.minus_eq
thf(fact_354_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    = ( ^ [A3: int,B2: int] :
          ( ( minus_minus_int @ A3 @ B2 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_355_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_356_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_357_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ( plus_plus_int @ C @ B )
        = A )
     => ( C
        = ( minus_minus_int @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_358_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_359_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_360_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_361_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_362_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_363_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A
        = ( minus_minus_int @ C @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_364_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( minus_minus_int @ A @ B )
        = C )
      = ( A
        = ( plus_plus_int @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_365_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_366_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_367_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( uminus_uminus_int @ A2 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_368_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_369_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_370_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_371_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_372_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_373_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_374_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_375_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% le_imp_neg_le
thf(fact_376_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_377_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% le_minus_iff
thf(fact_378_minus__diff__commute,axiom,
    ! [B: int,A: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
      = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_379_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_380_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_381_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_382_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_383_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_384_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_385_group__cancel_Osub2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3
        = ( plus_plus_int @ K @ B ) )
     => ( ( minus_minus_int @ A @ B3 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_386_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_387_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_388_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_389_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_390_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_391_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_392_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_393_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_394_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_395_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_396_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_397_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_398_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_399_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_400_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_401_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_402_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_403_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_404_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_405_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_406_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_407_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_408_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_409_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_mono
thf(fact_410_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_411_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_left_mono
thf(fact_412_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_413_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( B
           != ( plus_plus_nat @ A @ C2 ) ) ) ).

% less_eqE
thf(fact_414_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_right_mono
thf(fact_415_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_416_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B2: nat] :
        ? [C3: nat] :
          ( B2
          = ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_417_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_418_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_419_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_420_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_421_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_422_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_423_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_424_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_425_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] :
      ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
      = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_426_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_427_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_428_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_429_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_430_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_431_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_432_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_433_UNIV_I2_J,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ top_top_set_int ) ).

% UNIV(2)
thf(fact_434_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% add_eq_0_iff
thf(fact_435_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_group_add_class.ab_left_minus
thf(fact_436_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_437_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_438_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_439_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_440_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_441_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_442_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_443_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_444_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_445_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_446_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_447_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_448_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_449_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_450_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_451_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_452_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_453_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_454_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_455_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_456_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y @ zero_zero_int )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_457_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_458_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).

% of_nat_0_le_iff
thf(fact_459_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).

% of_nat_0_le_iff
thf(fact_460_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_461_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_462_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_463_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_464_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_465_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_466_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_467_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_468_int_Oadd_Oone__in__subset,axiom,
    ! [H: set_int] :
      ( ( ord_less_eq_set_int @ H @ top_top_set_int )
     => ( ( H != bot_bot_set_int )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ H )
             => ( member_int @ ( uminus_uminus_int @ X4 ) @ H ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ H )
               => ! [Xa: int] :
                    ( ( member_int @ Xa @ H )
                   => ( member_int @ ( plus_plus_int @ X4 @ Xa ) @ H ) ) )
           => ( member_int @ zero_zero_int @ H ) ) ) ) ) ).

% int.add.one_in_subset
thf(fact_469_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_470_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_471_square__diff__square__factored,axiom,
    ! [X: int,Y: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
      = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_472_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_473_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ ( ord_less_int @ A @ B )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_474_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_475_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_476_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_477_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_478_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_479_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_480_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_481_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_482_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_483_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_484_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_485_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_486_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_487_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_488_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_489_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_490_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_491_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_492_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_493_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_494_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_495_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_496_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_497_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_498_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_499_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_500_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_501_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_502_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_503_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_504_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_505_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_506_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_507_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_508_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_509_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_510_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_511_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_512_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_513_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_514_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_515_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_516_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_517_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_518_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_519_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_520_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_521_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_522_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_523_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_524_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_525_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_526_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_527_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_528_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_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_square__diff__one__factored,axiom,
    ! [X: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_532_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_533_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_534_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_535_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_536_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_537_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_538_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_539_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_540_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_541_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_542_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_543_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_544_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_545_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_546_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_547_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_548_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_549_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_550_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_551_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_552_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_553_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_554_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_555_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_556_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_557_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_558_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_559_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_560_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_561_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_562_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_563_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_564_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_565_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_566_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_567_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_568_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_569_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_570_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_571_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_572_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_573_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_int @ I @ K )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_574_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_575_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_576_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_577_group__cancel_Oadd2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B3 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_578_group__cancel_Oadd2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B3 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_579_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.assoc
thf(fact_580_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_581_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_582_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_583_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).

% add.commute
thf(fact_584_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).

% add.commute
thf(fact_585_add_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.left_commute
thf(fact_586_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_587_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_588_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_589_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_590_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_591_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_592_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_593_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.assoc
thf(fact_594_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_595_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).

% mult.commute
thf(fact_596_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).

% mult.commute
thf(fact_597_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_598_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_599_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_600_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_601_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_602_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_603_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
             => ( P @ M3 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_604_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_605_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_606_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_607_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_608_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_609_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_610_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_611_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_612_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_613_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_614_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_615_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_616_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_617_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_618_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_619_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_620_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_621_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_622_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_623_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_624_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_625_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P @ N2 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N2 )
                  & ~ ( P @ M3 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_626_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_627_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_628_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_629_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M
        = ( times_times_nat @ M @ N ) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_630_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_631_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_632_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_633_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_634_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_635_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_636_mult__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_637_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_638_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_639_int_Oadd_Oint__pow__diff,axiom,
    ! [X: int,N: int,M: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( minus_minus_int @ N @ M ) @ X )
        = ( plus_plus_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) @ ( uminus_uminus_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X ) ) ) ) ) ).

% int.add.int_pow_diff
thf(fact_640_int_Oa__coset__add__inv1,axiom,
    ! [M2: 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 ) ) ) @ M2 @ ( plus_plus_int @ X @ ( uminus_uminus_int @ Y ) ) )
        = M2 )
     => ( ( member_int @ X @ top_top_set_int )
       => ( ( member_int @ Y @ top_top_set_int )
         => ( ( ord_less_eq_set_int @ M2 @ 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 ) ) ) @ M2 @ 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 ) ) ) @ M2 @ Y ) ) ) ) ) ) ).

% int.a_coset_add_inv1
thf(fact_641_int_Oa__coset__add__inv2,axiom,
    ! [M2: 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 ) ) ) @ M2 @ 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 ) ) ) @ M2 @ Y ) )
     => ( ( member_int @ X @ top_top_set_int )
       => ( ( member_int @ Y @ top_top_set_int )
         => ( ( ord_less_eq_set_int @ M2 @ 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 ) ) ) @ M2 @ ( plus_plus_int @ X @ ( uminus_uminus_int @ Y ) ) )
              = M2 ) ) ) ) ) ).

% int.a_coset_add_inv2
thf(fact_642_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_643_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_644_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_645_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_646_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_647_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_648_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_649_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_650_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_651_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_652_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_653_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_654_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_655_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_656_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_657_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_658_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_659_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_660_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_661_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_662_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_663_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_664_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_665_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_666_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_667_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_668_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_669_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_670_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_671_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_672_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_673_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_674_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_675_int_Oadd_Oint__pow__inv,axiom,
    ! [X: int,I: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ ( uminus_uminus_int @ X ) )
        = ( uminus_uminus_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X ) ) ) ) ).

% int.add.int_pow_inv
thf(fact_676_int_Oadd_Oint__pow__neg,axiom,
    ! [X: int,I: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( uminus_uminus_int @ I ) @ X )
        = ( uminus_uminus_int @ ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I @ X ) ) ) ) ).

% int.add.int_pow_neg
thf(fact_677_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_678_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_679_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_680_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_681_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C2 ) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_682_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_683_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_684_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_685_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_686_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_687_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_688_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_689_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_690_int_Oa__rcosI,axiom,
    ! [H2: int,H: set_int,X: int] :
      ( ( member_int @ H2 @ H )
     => ( ( ord_less_eq_set_int @ H @ top_top_set_int )
       => ( ( member_int @ X @ top_top_set_int )
         => ( member_int @ ( plus_plus_int @ H2 @ X ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ H @ X ) ) ) ) ) ).

% int.a_rcosI
thf(fact_691_int_Oa__coset__add__assoc,axiom,
    ! [M2: set_int,G: int,H2: int] :
      ( ( ord_less_eq_set_int @ M2 @ top_top_set_int )
     => ( ( member_int @ G @ top_top_set_int )
       => ( ( member_int @ H2 @ top_top_set_int )
         => ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M2 @ G ) @ H2 )
            = ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M2 @ ( plus_plus_int @ G @ H2 ) ) ) ) ) ) ).

% int.a_coset_add_assoc
thf(fact_692_int_Oa__r__coset__subset__G,axiom,
    ! [H: set_int,X: int] :
      ( ( ord_less_eq_set_int @ H @ top_top_set_int )
     => ( ( member_int @ X @ top_top_set_int )
       => ( ord_less_eq_set_int @ ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ H @ X ) @ top_top_set_int ) ) ) ).

% int.a_r_coset_subset_G
thf(fact_693_int_Ogenideal__self,axiom,
    ! [S2: set_int] :
      ( ( ord_less_eq_set_int @ S2 @ top_top_set_int )
     => ( ord_less_eq_set_int @ S2 @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ S2 ) ) ) ).

% int.genideal_self
thf(fact_694_int_Osubset__Idl__subset,axiom,
    ! [I2: set_int,H: set_int] :
      ( ( ord_less_eq_set_int @ I2 @ top_top_set_int )
     => ( ( ord_less_eq_set_int @ H @ I2 )
       => ( ord_less_eq_set_int @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ H ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ I2 ) ) ) ) ).

% int.subset_Idl_subset
thf(fact_695_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_696_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_697_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% diff_numeral_special(12)
thf(fact_698_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_699_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_700_i_Oadd_Oint__pow__neg__int,axiom,
    ! [X: int,N: nat] :
      ( ( 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_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ 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 ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) ) ) ) ).

% i.add.int_pow_neg_int
thf(fact_701_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
        = ( semiri1314217659103216013at_int @ M ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_702_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( ( insert_int @ B @ bot_bot_set_int )
        = ( insert_int @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_703_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( ( insert_int @ A @ A2 )
        = ( insert_int @ B @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_704_boolean__algebra_Ocompl__one,axiom,
    ( ( uminus_uminus_int_o @ top_top_int_o )
    = bot_bot_int_o ) ).

% boolean_algebra.compl_one
thf(fact_705_boolean__algebra_Ocompl__one,axiom,
    ( ( uminus1532241313380277803et_int @ top_top_set_int )
    = bot_bot_set_int ) ).

% boolean_algebra.compl_one
thf(fact_706_UNIV__I,axiom,
    ! [X: int] : ( member_int @ X @ top_top_set_int ) ).

% UNIV_I
thf(fact_707_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_708_int_Ole__refl,axiom,
    ! [X: int] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ord_less_eq_int @ X @ X ) ) ).

% int.le_refl
thf(fact_709_Compl__anti__mono,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B3 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).

% Compl_anti_mono
thf(fact_710_Compl__subset__Compl__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B3 ) )
      = ( ord_less_eq_set_int @ B3 @ A2 ) ) ).

% Compl_subset_Compl_iff
thf(fact_711_subsetI,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( member_int @ X4 @ B3 ) )
     => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).

% subsetI
thf(fact_712_subset__antisym,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ A2 )
       => ( A2 = B3 ) ) ) ).

% subset_antisym
thf(fact_713_psubsetI,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( A2 != B3 )
       => ( ord_less_set_int @ A2 @ B3 ) ) ) ).

% psubsetI
thf(fact_714_Diff__cancel,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ A2 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_715_empty__Diff,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_716_Diff__empty,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
      = A2 ) ).

% Diff_empty
thf(fact_717_empty__Collect__eq,axiom,
    ! [P: int > $o] :
      ( ( bot_bot_set_int
        = ( collect_int @ P ) )
      = ( ! [X2: int] :
            ~ ( P @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_718_Collect__empty__eq,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( ! [X2: int] :
            ~ ( P @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_719_all__not__in__conv,axiom,
    ! [A2: set_int] :
      ( ( ! [X2: int] :
            ~ ( member_int @ X2 @ A2 ) )
      = ( A2 = bot_bot_set_int ) ) ).

% all_not_in_conv
thf(fact_720_empty__iff,axiom,
    ! [C: int] :
      ~ ( member_int @ C @ bot_bot_set_int ) ).

% empty_iff
thf(fact_721_insert__Diff1,axiom,
    ! [X: int,B3: set_int,A2: set_int] :
      ( ( member_int @ X @ B3 )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B3 )
        = ( minus_minus_set_int @ A2 @ B3 ) ) ) ).

% insert_Diff1
thf(fact_722_Diff__insert0,axiom,
    ! [X: int,A2: set_int,B3: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( minus_minus_set_int @ A2 @ ( insert_int @ X @ B3 ) )
        = ( minus_minus_set_int @ A2 @ B3 ) ) ) ).

% Diff_insert0
thf(fact_723_insert__absorb2,axiom,
    ! [X: int,A2: set_int] :
      ( ( insert_int @ X @ ( insert_int @ X @ A2 ) )
      = ( insert_int @ X @ A2 ) ) ).

% insert_absorb2
thf(fact_724_insert__iff,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
      = ( ( A = B )
        | ( member_int @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_725_insertCI,axiom,
    ! [A: int,B3: set_int,B: int] :
      ( ( ~ ( member_int @ A @ B3 )
       => ( A = B ) )
     => ( member_int @ A @ ( insert_int @ B @ B3 ) ) ) ).

% insertCI
thf(fact_726_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [X: set_int] :
      ( ( uminus1532241313380277803et_int @ ( uminus1532241313380277803et_int @ X ) )
      = X ) ).

% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_727_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( uminus1532241313380277803et_int @ X )
        = ( uminus1532241313380277803et_int @ Y ) )
      = ( X = Y ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_728_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_729_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_730_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_731_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_732_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_733_Diff__UNIV,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ top_top_set_int )
      = bot_bot_set_int ) ).

% Diff_UNIV
thf(fact_734_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_735_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ( minus_minus_set_int @ A2 @ B3 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A2 @ B3 ) ) ).

% Diff_eq_empty_iff
thf(fact_736_empty__subsetI,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).

% empty_subsetI
thf(fact_737_subset__empty,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
      = ( A2 = bot_bot_set_int ) ) ).

% subset_empty
thf(fact_738_compl__less__compl__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y ) )
      = ( ord_less_set_int @ Y @ X ) ) ).

% compl_less_compl_iff
thf(fact_739_insert__subset,axiom,
    ! [X: int,A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int @ X @ A2 ) @ B3 )
      = ( ( member_int @ X @ B3 )
        & ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ).

% insert_subset
thf(fact_740_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] :
      ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
      = ( insert_int @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_741_singletonI,axiom,
    ! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).

% singletonI
thf(fact_742_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_743_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_744_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_745_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_746_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_747_negative__zle,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_748_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_749_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_750_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_751_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_752_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_753_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_754_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_755_i_Oadd_Opow__mult__distrib,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( 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 ) )
     => ( ( 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_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ ( 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 ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ Y ) ) ) ) ) ) ).

% i.add.pow_mult_distrib
thf(fact_756_i_Oadd_Onat__pow__distrib,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( 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_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ ( 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 ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ Y ) ) ) ) ) ).

% i.add.nat_pow_distrib
thf(fact_757_i_Oadd_Onat__pow__comm,axiom,
    ! [X: int,N: nat,M: nat] :
      ( ( 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 ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X ) )
        = ( 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_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) ) ) ) ).

% i.add.nat_pow_comm
thf(fact_758_i_Oadd_Ogroup__commutes__pow,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( 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 ) )
     => ( ( 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 ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ 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 @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) ) ) ) ) ) ).

% i.add.group_commutes_pow
thf(fact_759_i_Oadd_Opow__eq__div2,axiom,
    ! [X: int,M: nat,N: nat] :
      ( ( 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_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X )
          = ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) )
       => ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( minus_minus_nat @ M @ N ) @ 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.pow_eq_div2
thf(fact_760_boolean__algebra_Ocompl__zero,axiom,
    ( ( uminus_uminus_int_o @ bot_bot_int_o )
    = top_top_int_o ) ).

% boolean_algebra.compl_zero
thf(fact_761_boolean__algebra_Ocompl__zero,axiom,
    ( ( uminus1532241313380277803et_int @ bot_bot_set_int )
    = top_top_set_int ) ).

% boolean_algebra.compl_zero
thf(fact_762_i_Oadd_Onat__pow__pow,axiom,
    ! [X: int,M: nat,N: nat] :
      ( ( 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_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) )
        = ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( times_times_nat @ N @ M ) @ X ) ) ) ).

% i.add.nat_pow_pow
thf(fact_763_subset__Compl__singleton,axiom,
    ! [A2: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
      = ( ~ ( member_int @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_764_i_Oadd_Onat__pow__inv,axiom,
    ! [X: int,I: nat] :
      ( ( 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_po6256809715947028093it_nat @ ( partia4118392927963588428t_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_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 ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_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.add.nat_pow_inv
thf(fact_765_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_766_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
      = ( ord_less_int @ W @ Z ) ) ).

% zle_diff1_eq
thf(fact_767_i_Oadd_Onat__pow__mult,axiom,
    ! [X: int,N: nat,M: nat] :
      ( ( 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 ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X ) )
        = ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_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_nat @ N @ M ) @ X ) ) ) ).

% i.add.nat_pow_mult
thf(fact_768_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% mult_le_cancel2
thf(fact_769_int_Oadd_Onat__pow__closed,axiom,
    ! [X: int,N: nat] :
      ( ( member_int @ X @ top_top_set_int )
     => ( member_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) @ top_top_set_int ) ) ).

% int.add.nat_pow_closed
thf(fact_770_int_Oadd_Onat__pow__one,axiom,
    ! [N: nat] :
      ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ zero_zero_int )
      = zero_zero_int ) ).

% int.add.nat_pow_one
thf(fact_771_i_Oadd_Onat__pow__closed,axiom,
    ! [X: int,N: nat] :
      ( ( 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 @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ 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.nat_pow_closed
thf(fact_772_int_Oadd_Onat__pow__0,axiom,
    ! [X: int] :
      ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_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_nat @ X )
      = zero_zero_int ) ).

% int.add.nat_pow_0
thf(fact_773_i_Oadd_Onat__pow__one,axiom,
    ! [N: nat] :
      ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ ( zero_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_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.nat_pow_one
thf(fact_774_i_Oadd_Onat__pow__0,axiom,
    ! [X: int] :
      ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_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_nat @ 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.nat_pow_0
thf(fact_775_i_Oadd_Onat__pow__eone,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_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ one_one_nat @ X )
        = X ) ) ).

% i.add.nat_pow_eone
thf(fact_776_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_777_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_778_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_779_int_Olless__eq,axiom,
    ( ord_less_int
    = ( ^ [X2: int,Y3: int] :
          ( ( ord_less_eq_int @ X2 @ Y3 )
          & ( X2 != Y3 ) ) ) ) ).

% int.lless_eq
thf(fact_780_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y5: nat] :
            ( ( P @ Y5 )
           => ( ord_less_eq_nat @ Y5 @ B ) )
       => ? [X4: nat] :
            ( ( P @ X4 )
            & ! [Y6: nat] :
                ( ( P @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_781_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_782_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_783_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_784_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_785_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_786_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_787_double__diff,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ C4 )
       => ( ( minus_minus_set_int @ B3 @ ( minus_minus_set_int @ C4 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_788_Diff__subset,axiom,
    ! [A2: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ A2 ) ).

% Diff_subset
thf(fact_789_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_790_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_791_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_792_Diff__mono,axiom,
    ! [A2: set_int,C4: set_int,D2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ C4 )
     => ( ( ord_less_eq_set_int @ D2 @ B3 )
       => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ ( minus_minus_set_int @ C4 @ D2 ) ) ) ) ).

% Diff_mono
thf(fact_793_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_794_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_795_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_796_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% zle_int
thf(fact_797_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_798_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_799_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_800_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_801_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_802_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_803_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N2: nat] :
          ( L
          = ( plus_plus_nat @ K @ N2 ) ) ) ).

% le_Suc_ex
thf(fact_804_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_805_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_806_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_807_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_808_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_809_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_810_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_811_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_812_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_813_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_814_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_815_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_816_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_817_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M4: nat,N3: nat] :
        ? [K2: nat] :
          ( N3
          = ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).

% nat_le_iff_add
thf(fact_818_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_819_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% add_mult_distrib
thf(fact_820_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_821_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% add_mult_distrib2
thf(fact_822_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_823_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% diff_mult_distrib
thf(fact_824_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% diff_mult_distrib2
thf(fact_825_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_826_Compl__eq__Diff__UNIV,axiom,
    ( uminus1532241313380277803et_int
    = ( minus_minus_set_int @ top_top_set_int ) ) ).

% Compl_eq_Diff_UNIV
thf(fact_827_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_828_insert__Diff__if,axiom,
    ! [X: int,B3: set_int,A2: set_int] :
      ( ( ( member_int @ X @ B3 )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B3 )
          = ( minus_minus_set_int @ A2 @ B3 ) ) )
      & ( ~ ( member_int @ X @ B3 )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B3 )
          = ( insert_int @ X @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_829_bot__set__def,axiom,
    ( bot_bot_set_int
    = ( collect_int @ bot_bot_int_o ) ) ).

% bot_set_def
thf(fact_830_top__set__def,axiom,
    ( top_top_set_int
    = ( collect_int @ top_top_int_o ) ) ).

% top_set_def
thf(fact_831_Compl__insert,axiom,
    ! [X: int,A2: set_int] :
      ( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A2 ) )
      = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% Compl_insert
thf(fact_832_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_833_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_834_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_835_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_836_diff__add__0,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_837_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_838_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_839_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_840_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_841_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_842_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_843_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_844_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_845_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_846_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M5: nat,N2: nat] :
          ( ( ord_less_nat @ M5 @ N2 )
         => ( ord_less_nat @ ( F @ M5 ) @ ( F @ N2 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_847_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
          & ( M4 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_848_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_849_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M4: nat,N3: nat] :
          ( ( ord_less_nat @ M4 @ N3 )
          | ( M4 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_850_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_851_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_852_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_nat @ I3 @ J2 )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_853_Compl__empty__eq,axiom,
    ( ( uminus1532241313380277803et_int @ bot_bot_set_int )
    = top_top_set_int ) ).

% Compl_empty_eq
thf(fact_854_Compl__UNIV__eq,axiom,
    ( ( uminus1532241313380277803et_int @ top_top_set_int )
    = bot_bot_set_int ) ).

% Compl_UNIV_eq
thf(fact_855_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
      = ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
thf(fact_856_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_857_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nonneg_int_cases
thf(fact_858_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W2: int,Z3: int] :
        ? [N3: nat] :
          ( Z3
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_859_subset__Diff__insert,axiom,
    ! [A2: set_int,B3: set_int,X: int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B3 @ ( insert_int @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B3 @ C4 ) )
        & ~ ( member_int @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_860_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_861_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_862_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_863_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_864_Diff__insert__absorb,axiom,
    ! [X: int,A2: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_865_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B3: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B3 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B3 ) ) ).

% Diff_insert2
thf(fact_866_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_867_Diff__insert,axiom,
    ! [A2: set_int,A: int,B3: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B3 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_868_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_869_Diff__single__insert,axiom,
    ! [A2: set_int,X: int,B3: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B3 )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B3 ) ) ) ).

% Diff_single_insert
thf(fact_870_subset__insert__iff,axiom,
    ! [A2: set_int,X: int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B3 ) )
      = ( ( ( member_int @ X @ A2 )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B3 ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ) ).

% subset_insert_iff
thf(fact_871_psubset__insert__iff,axiom,
    ! [A2: set_int,X: int,B3: set_int] :
      ( ( ord_less_set_int @ A2 @ ( insert_int @ X @ B3 ) )
      = ( ( ( member_int @ X @ B3 )
         => ( ord_less_set_int @ A2 @ B3 ) )
        & ( ~ ( member_int @ X @ B3 )
         => ( ( ( member_int @ X @ A2 )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B3 ) )
            & ( ~ ( member_int @ X @ A2 )
             => ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_872_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_873_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% nonpos_int_cases
thf(fact_874_negative__zle__0,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_875_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_876_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_877_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% of_nat_diff
thf(fact_878_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I3: int] :
            ( ( ord_less_eq_int @ K @ I3 )
           => ( ( P @ I3 )
             => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_879_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D3: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D3 ) )
                & ~ ( P @ D3 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_880_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D3: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D3 ) )
           => ( P @ D3 ) ) ) ) ).

% nat_diff_split
thf(fact_881_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_882_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% mult_eq_if
thf(fact_883_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K3 )
               => ~ ( P @ I4 ) )
            & ( P @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_884_UNIV__witness,axiom,
    ? [X4: int] : ( member_int @ X4 @ top_top_set_int ) ).

% UNIV_witness
thf(fact_885_UNIV__eq__I,axiom,
    ! [A2: set_int] :
      ( ! [X4: int] : ( member_int @ X4 @ A2 )
     => ( top_top_set_int = A2 ) ) ).

% UNIV_eq_I
thf(fact_886_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_887_int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% int_int_eq
thf(fact_888_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_889_in__mono,axiom,
    ! [A2: set_int,B3: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( member_int @ X @ A2 )
       => ( member_int @ X @ B3 ) ) ) ).

% in_mono
thf(fact_890_subsetD,axiom,
    ! [A2: set_int,B3: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( member_int @ C @ A2 )
       => ( member_int @ C @ B3 ) ) ) ).

% subsetD
thf(fact_891_equalityE,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B3 )
         => ~ ( ord_less_eq_set_int @ B3 @ A2 ) ) ) ).

% equalityE
thf(fact_892_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A4: set_int,B4: set_int] :
        ! [X2: int] :
          ( ( member_int @ X2 @ A4 )
         => ( member_int @ X2 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_893_equalityD1,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).

% equalityD1
thf(fact_894_equalityD2,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => ( ord_less_eq_set_int @ B3 @ A2 ) ) ).

% equalityD2
thf(fact_895_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A4: set_int,B4: set_int] :
        ! [T2: int] :
          ( ( member_int @ T2 @ A4 )
         => ( member_int @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_896_subset__refl,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).

% subset_refl
thf(fact_897_Collect__mono,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_mono
thf(fact_898_subset__trans,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ C4 )
       => ( ord_less_eq_set_int @ A2 @ C4 ) ) ) ).

% subset_trans
thf(fact_899_set__eq__subset,axiom,
    ( ( ^ [Y4: set_int,Z2: set_int] : ( Y4 = Z2 ) )
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ A4 @ B4 )
          & ( ord_less_eq_set_int @ B4 @ A4 ) ) ) ) ).

% set_eq_subset
thf(fact_900_Collect__mono__iff,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
      = ( ! [X2: int] :
            ( ( P @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_901_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( ord_less_set_int @ A4 @ B4 )
          | ( A4 = B4 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_902_subset__psubset__trans,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( ord_less_set_int @ B3 @ C4 )
       => ( ord_less_set_int @ A2 @ C4 ) ) ) ).

% subset_psubset_trans
thf(fact_903_subset__not__subset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ A4 @ B4 )
          & ~ ( ord_less_eq_set_int @ B4 @ A4 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_904_psubset__subset__trans,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ C4 )
       => ( ord_less_set_int @ A2 @ C4 ) ) ) ).

% psubset_subset_trans
thf(fact_905_psubset__imp__subset,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).

% psubset_imp_subset
thf(fact_906_psubset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A4: set_int,B4: set_int] :
          ( ( ord_less_eq_set_int @ A4 @ B4 )
          & ( A4 != B4 ) ) ) ) ).

% psubset_eq
thf(fact_907_psubsetE,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B3 )
         => ( ord_less_eq_set_int @ B3 @ A2 ) ) ) ).

% psubsetE
thf(fact_908_ex__in__conv,axiom,
    ! [A2: set_int] :
      ( ( ? [X2: int] : ( member_int @ X2 @ A2 ) )
      = ( A2 != bot_bot_set_int ) ) ).

% ex_in_conv
thf(fact_909_equals0I,axiom,
    ! [A2: set_int] :
      ( ! [Y5: int] :
          ~ ( member_int @ Y5 @ A2 )
     => ( A2 = bot_bot_set_int ) ) ).

% equals0I
thf(fact_910_equals0D,axiom,
    ! [A2: set_int,A: int] :
      ( ( A2 = bot_bot_set_int )
     => ~ ( member_int @ A @ A2 ) ) ).

% equals0D
thf(fact_911_emptyE,axiom,
    ! [A: int] :
      ~ ( member_int @ A @ bot_bot_set_int ) ).

% emptyE
thf(fact_912_not__psubset__empty,axiom,
    ! [A2: set_int] :
      ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_913_mk__disjoint__insert,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ? [B5: set_int] :
          ( ( A2
            = ( insert_int @ A @ B5 ) )
          & ~ ( member_int @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_914_insert__commute,axiom,
    ! [X: int,Y: int,A2: set_int] :
      ( ( insert_int @ X @ ( insert_int @ Y @ A2 ) )
      = ( insert_int @ Y @ ( insert_int @ X @ A2 ) ) ) ).

% insert_commute
thf(fact_915_insert__eq__iff,axiom,
    ! [A: int,A2: set_int,B: int,B3: set_int] :
      ( ~ ( member_int @ A @ A2 )
     => ( ~ ( member_int @ B @ B3 )
       => ( ( ( insert_int @ A @ A2 )
            = ( insert_int @ B @ B3 ) )
          = ( ( ( A = B )
             => ( A2 = B3 ) )
            & ( ( A != B )
             => ? [C5: set_int] :
                  ( ( A2
                    = ( insert_int @ B @ C5 ) )
                  & ~ ( member_int @ B @ C5 )
                  & ( B3
                    = ( insert_int @ A @ C5 ) )
                  & ~ ( member_int @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_916_insert__absorb,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ( ( insert_int @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_917_insert__ident,axiom,
    ! [X: int,A2: set_int,B3: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ~ ( member_int @ X @ B3 )
       => ( ( ( insert_int @ X @ A2 )
            = ( insert_int @ X @ B3 ) )
          = ( A2 = B3 ) ) ) ) ).

% insert_ident
thf(fact_918_Set_Oset__insert,axiom,
    ! [X: int,A2: set_int] :
      ( ( member_int @ X @ A2 )
     => ~ ! [B5: set_int] :
            ( ( A2
              = ( insert_int @ X @ B5 ) )
           => ( member_int @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_919_insertI2,axiom,
    ! [A: int,B3: set_int,B: int] :
      ( ( member_int @ A @ B3 )
     => ( member_int @ A @ ( insert_int @ B @ B3 ) ) ) ).

% insertI2
thf(fact_920_insertI1,axiom,
    ! [A: int,B3: set_int] : ( member_int @ A @ ( insert_int @ A @ B3 ) ) ).

% insertI1
thf(fact_921_insertE,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
     => ( ( A != B )
       => ( member_int @ A @ A2 ) ) ) ).

% insertE
thf(fact_922_int_Oadd_Opow__eq__div2,axiom,
    ! [X: int,M: nat,N: nat] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X )
          = ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) )
       => ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( minus_minus_nat @ M @ N ) @ X )
          = zero_zero_int ) ) ) ).

% int.add.pow_eq_div2
thf(fact_923_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_924_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_925_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_926_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_927_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_928_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_929_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).

% zadd_int_left
thf(fact_930_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_931_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_932_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus_int @ K @ zero_zero_int )
      = K ) ).

% plus_int_code(1)
thf(fact_933_plus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( plus_plus_int @ zero_zero_int @ L )
      = L ) ).

% plus_int_code(2)
thf(fact_934_subset__UNIV,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ top_top_set_int ) ).

% subset_UNIV
thf(fact_935_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_936_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_937_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_938_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_939_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_940_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_941_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_942_empty__not__UNIV,axiom,
    bot_bot_set_int != top_top_set_int ).

% empty_not_UNIV
thf(fact_943_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_944_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_945_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_946_insert__UNIV,axiom,
    ! [X: int] :
      ( ( insert_int @ X @ top_top_set_int )
      = top_top_set_int ) ).

% insert_UNIV
thf(fact_947_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_948_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_949_compl__less__swap2,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_set_int @ ( uminus1532241313380277803et_int @ Y ) @ X )
     => ( ord_less_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) ) ).

% compl_less_swap2
thf(fact_950_compl__less__swap1,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) )
     => ( ord_less_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).

% compl_less_swap1
thf(fact_951_subset__insertI2,axiom,
    ! [A2: set_int,B3: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ B3 ) ) ) ).

% subset_insertI2
thf(fact_952_subset__insertI,axiom,
    ! [B3: set_int,A: int] : ( ord_less_eq_set_int @ B3 @ ( insert_int @ A @ B3 ) ) ).

% subset_insertI
thf(fact_953_subset__insert,axiom,
    ! [X: int,A2: set_int,B3: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B3 ) )
        = ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ).

% subset_insert
thf(fact_954_insert__mono,axiom,
    ! [C4: set_int,D2: set_int,A: int] :
      ( ( ord_less_eq_set_int @ C4 @ D2 )
     => ( ord_less_eq_set_int @ ( insert_int @ A @ C4 ) @ ( insert_int @ A @ D2 ) ) ) ).

% insert_mono
thf(fact_955_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( ( insert_int @ A @ bot_bot_set_int )
        = ( insert_int @ B @ bot_bot_set_int ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_956_insert__not__empty,axiom,
    ! [A: int,A2: set_int] :
      ( ( insert_int @ A @ A2 )
     != bot_bot_set_int ) ).

% insert_not_empty
thf(fact_957_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
        = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_958_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_959_singletonD,axiom,
    ! [B: int,A: int] :
      ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_960_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N2: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% int_cases2
thf(fact_961_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% not_int_zless_negative
thf(fact_962_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M5: nat,N2: nat] :
          ( Z
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% int_diff_cases
thf(fact_963_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_964_int__less__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_less_induct
thf(fact_965_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
      = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(4)
thf(fact_966_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
      = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(3)
thf(fact_967_int_Oadd_Ogroup__commutes__pow,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( plus_plus_int @ X @ Y )
        = ( plus_plus_int @ Y @ X ) )
     => ( ( member_int @ X @ top_top_set_int )
       => ( ( member_int @ Y @ top_top_set_int )
         => ( ( plus_plus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) @ Y )
            = ( plus_plus_int @ Y @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) ) ) ) ) ) ).

% int.add.group_commutes_pow
thf(fact_968_int_Oadd_Opow__mult__distrib,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( plus_plus_int @ X @ Y )
        = ( plus_plus_int @ Y @ X ) )
     => ( ( member_int @ X @ top_top_set_int )
       => ( ( member_int @ Y @ top_top_set_int )
         => ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ ( plus_plus_int @ X @ Y ) )
            = ( plus_plus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ Y ) ) ) ) ) ) ).

% int.add.pow_mult_distrib
thf(fact_969_int_Oadd_Onat__pow__distrib,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( member_int @ Y @ top_top_set_int )
       => ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ ( plus_plus_int @ X @ Y ) )
          = ( plus_plus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ Y ) ) ) ) ) ).

% int.add.nat_pow_distrib
thf(fact_970_int_Oadd_Onat__pow__comm,axiom,
    ! [X: int,N: nat,M: nat] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( plus_plus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X ) )
        = ( plus_plus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) ) ) ) ).

% int.add.nat_pow_comm
thf(fact_971_int_Oadd__pow__rdistr,axiom,
    ! [A: int,B: int,K: nat] :
      ( ( member_int @ A @ top_top_set_int )
     => ( ( member_int @ B @ top_top_set_int )
       => ( ( times_times_int @ A @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K @ B ) )
          = ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K @ ( times_times_int @ A @ B ) ) ) ) ) ).

% int.add_pow_rdistr
thf(fact_972_int_Oadd__pow__ldistr,axiom,
    ! [A: int,B: int,K: nat] :
      ( ( member_int @ A @ top_top_set_int )
     => ( ( member_int @ B @ top_top_set_int )
       => ( ( times_times_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K @ A ) @ B )
          = ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ K @ ( times_times_int @ A @ B ) ) ) ) ) ).

% int.add_pow_ldistr
thf(fact_973_int_Oadd_Onat__pow__pow,axiom,
    ! [X: int,M: nat,N: nat] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) )
        = ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( times_times_nat @ N @ M ) @ X ) ) ) ).

% int.add.nat_pow_pow
thf(fact_974_int_Oadd_Onat__pow__mult,axiom,
    ! [X: int,N: nat,M: nat] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( plus_plus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ M @ X ) )
        = ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_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_nat @ N @ M ) @ X ) ) ) ).

% int.add.nat_pow_mult
thf(fact_975_int_Oadd_Onat__pow__inv,axiom,
    ! [X: int,I: nat] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_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 ) )
        = ( uminus_uminus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_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 ) ) ) ) ).

% int.add.nat_pow_inv
thf(fact_976_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_977_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_978_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_979_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_980_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_981_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_982_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_983_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_984_diff__shunt__var,axiom,
    ! [X: int > $o,Y: int > $o] :
      ( ( ( minus_minus_int_o @ X @ Y )
        = bot_bot_int_o )
      = ( ord_less_eq_int_o @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_985_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( minus_minus_set_int @ X @ Y )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_986_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_987_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_988_subset__singleton__iff,axiom,
    ! [X5: set_int,A: int] :
      ( ( ord_less_eq_set_int @ X5 @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( ( X5 = bot_bot_set_int )
        | ( X5
          = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_989_subset__singletonD,axiom,
    ! [A2: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) )
     => ( ( A2 = bot_bot_set_int )
        | ( A2
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% subset_singletonD
thf(fact_990_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( ( times_times_int @ M @ N )
        = one_one_int )
     => ( ( M = one_one_int )
        | ( M
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_991_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( ( times_times_int @ M @ N )
        = one_one_int )
      = ( ( ( M = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_992_minus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( minus_minus_int @ zero_zero_int @ L )
      = ( uminus_uminus_int @ L ) ) ).

% minus_int_code(2)
thf(fact_993_int_Oadd_Oint__pow__neg__int,axiom,
    ! [X: int,N: nat] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( add_po6254319245437977817it_int @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ X )
        = ( uminus_uminus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) ) ) ) ).

% int.add.int_pow_neg_int
thf(fact_994_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_995_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_996_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_997_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_998_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N2 )
          & ( K
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_999_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% pos_int_cases
thf(fact_1000_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_1001_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% int_cases4
thf(fact_1002_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
       => ~ ! [N2: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% int_cases3
thf(fact_1003_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% neg_int_cases
thf(fact_1004_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_1005_i_Oadd_Onat__pow__Suc2,axiom,
    ! [X: int,N: nat] :
      ( ( 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_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( suc @ N ) @ 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 @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) ) ) ) ).

% i.add.nat_pow_Suc2
thf(fact_1006_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_1007_verit__minus__simplify_I3_J,axiom,
    ! [B: int] :
      ( ( minus_minus_int @ zero_zero_int @ B )
      = ( uminus_uminus_int @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_1008_verit__minus__simplify_I4_J,axiom,
    ! [B: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_1009_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_1010_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_1011_Diff__idemp,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ B3 )
      = ( minus_minus_set_int @ A2 @ B3 ) ) ).

% Diff_idemp
thf(fact_1012_Diff__iff,axiom,
    ! [C: int,A2: set_int,B3: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
      = ( ( member_int @ C @ A2 )
        & ~ ( member_int @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_1013_DiffI,axiom,
    ! [C: int,A2: set_int,B3: set_int] :
      ( ( member_int @ C @ A2 )
     => ( ~ ( member_int @ C @ B3 )
       => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ) ).

% DiffI
thf(fact_1014_Compl__eq__Compl__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ( uminus1532241313380277803et_int @ A2 )
        = ( uminus1532241313380277803et_int @ B3 ) )
      = ( A2 = B3 ) ) ).

% Compl_eq_Compl_iff
thf(fact_1015_Compl__iff,axiom,
    ! [C: int,A2: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
      = ( ~ ( member_int @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_1016_ComplI,axiom,
    ! [C: int,A2: set_int] :
      ( ~ ( member_int @ C @ A2 )
     => ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).

% ComplI
thf(fact_1017_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_1018_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_1019_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_1020_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_1021_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_1022_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_Suc_Suc
thf(fact_1023_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_1024_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1025_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1026_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_1027_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_1028_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_1029_mult__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ M @ ( suc @ N ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc_right
thf(fact_1030_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_1031_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_1032_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_1033_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).

% one_le_mult_iff
thf(fact_1034_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_1035_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_1036_negative__zless,axiom,
    ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_1037_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_1038_int_Oadd_Onat__pow__Suc,axiom,
    ! [N: nat,X: int] :
      ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( suc @ N ) @ X )
      = ( plus_plus_int @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) @ X ) ) ).

% int.add.nat_pow_Suc
thf(fact_1039_i_Oadd_Onat__pow__Suc,axiom,
    ! [N: nat,X: int] :
      ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( suc @ N ) @ 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 ) ) ) @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) @ X ) ) ).

% i.add.nat_pow_Suc
thf(fact_1040_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_1041_psubsetD,axiom,
    ! [A2: set_int,B3: set_int,C: int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ( ( member_int @ C @ A2 )
       => ( member_int @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_1042_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_1043_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_1044_psubset__trans,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ( ( ord_less_set_int @ B3 @ C4 )
       => ( ord_less_set_int @ A2 @ C4 ) ) ) ).

% psubset_trans
thf(fact_1045_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ? [B6: int] : ( member_int @ B6 @ ( minus_minus_set_int @ B3 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1046_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R2: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X4: nat] : ( R2 @ X4 @ X4 )
       => ( ! [X4: nat,Y5: nat,Z4: nat] :
              ( ( R2 @ X4 @ Y5 )
             => ( ( R2 @ Y5 @ Z4 )
               => ( R2 @ X4 @ Z4 ) ) )
         => ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
           => ( R2 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1047_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1048_double__complement,axiom,
    ! [A2: set_int] :
      ( ( uminus1532241313380277803et_int @ ( uminus1532241313380277803et_int @ A2 ) )
      = A2 ) ).

% double_complement
thf(fact_1049_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_1050_verit__la__generic,axiom,
    ! [A: int,X: int] :
      ( ( ord_less_eq_int @ A @ X )
      | ( A = X )
      | ( ord_less_eq_int @ X @ A ) ) ).

% verit_la_generic
thf(fact_1051_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
             => ( P @ M3 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_1052_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_1053_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1054_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).

% diff_commute
thf(fact_1055_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1056_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_1057_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
     => ? [M5: nat] :
          ( M6
          = ( suc @ M5 ) ) ) ).

% Suc_le_D
thf(fact_1058_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1059_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_1060_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_1061_DiffD2,axiom,
    ! [C: int,A2: set_int,B3: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
     => ~ ( member_int @ C @ B3 ) ) ).

% DiffD2
thf(fact_1062_DiffD1,axiom,
    ! [C: int,A2: set_int,B3: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
     => ( member_int @ C @ A2 ) ) ).

% DiffD1
thf(fact_1063_ComplD,axiom,
    ! [C: int,A2: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
     => ~ ( member_int @ C @ A2 ) ) ).

% ComplD
thf(fact_1064_DiffE,axiom,
    ! [C: int,A2: set_int,B3: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
     => ~ ( ( member_int @ C @ A2 )
         => ( member_int @ C @ B3 ) ) ) ).

% DiffE
thf(fact_1065_nat__arith_Osuc1,axiom,
    ! [A2: nat,K: nat,A: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( suc @ A2 )
        = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_1066_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_1067_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_1068_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( M = N ) ) ).

% Suc_mult_cancel1
thf(fact_1069_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M5: nat] :
          ( N
          = ( suc @ M5 ) ) ) ).

% not0_implies_Suc
thf(fact_1070_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_1071_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_1072_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1073_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1074_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
     => ( ! [Y5: nat] : ( P @ zero_zero_nat @ ( suc @ Y5 ) )
       => ( ! [X4: nat,Y5: nat] :
              ( ( P @ X4 @ Y5 )
             => ( P @ ( suc @ X4 ) @ ( suc @ Y5 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_1075_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_1076_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1077_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1078_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1079_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1080_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_1081_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_1082_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_1083_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_1084_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_1085_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_1086_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_1087_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( suc @ N ) )
            & ( P @ I5 ) ) )
      = ( ( P @ N )
        | ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ N )
            & ( P @ I5 ) ) ) ) ).

% Ex_less_Suc
thf(fact_1088_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_1089_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_1090_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( suc @ N ) )
           => ( P @ I5 ) ) )
      = ( ( P @ N )
        & ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ N )
           => ( P @ I5 ) ) ) ) ).

% All_less_Suc
thf(fact_1091_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M7: nat] :
            ( ( M
              = ( suc @ M7 ) )
            & ( ord_less_nat @ N @ M7 ) ) ) ) ).

% Suc_less_eq2
thf(fact_1092_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_1093_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_1094_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_1095_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J2: nat,K3: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( ( ord_less_nat @ J2 @ K3 )
               => ( ( P @ I3 @ J2 )
                 => ( ( P @ J2 @ K3 )
                   => ( P @ I3 @ K3 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_1096_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] :
            ( ( J
              = ( suc @ I3 ) )
           => ( P @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J )
             => ( ( P @ ( suc @ I3 ) )
               => ( P @ I3 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_1097_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_1098_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_1099_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_1100_int__ops_I4_J,axiom,
    ! [A: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).

% int_ops(4)
thf(fact_1101_int__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).

% int_Suc
thf(fact_1102_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_1103_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_1104_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_set_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1105_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1106_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1107_lift__Suc__mono__le,axiom,
    ! [F: nat > set_int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1108_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1109_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1110_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > set_int,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_set_int @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1111_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1112_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1113_lift__Suc__mono__less,axiom,
    ! [F: nat > set_int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_set_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_set_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1114_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1115_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1116_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( suc @ N ) )
            & ( P @ I5 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ N )
            & ( P @ ( suc @ I5 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_1117_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M4: nat] :
            ( N
            = ( suc @ M4 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1118_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( suc @ N ) )
           => ( P @ I5 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ N )
           => ( P @ ( suc @ I5 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_1119_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M5: nat] :
          ( N
          = ( suc @ M5 ) ) ) ).

% gr0_implies_Suc
thf(fact_1120_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1121_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_1122_int__of__nat__induct,axiom,
    ! [P: int > $o,Z: int] :
      ( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
       => ( P @ Z ) ) ) ).

% int_of_nat_induct
thf(fact_1123_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% int_cases
thf(fact_1124_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1125_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1126_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_1127_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_1128_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_1129_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_1130_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_1131_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_1132_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_1133_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_1134_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_1135_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_1136_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q2: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).

% less_natE
thf(fact_1137_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_1138_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_1139_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M4: nat,N3: nat] :
        ? [K2: nat] :
          ( N3
          = ( suc @ ( plus_plus_nat @ M4 @ K2 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1140_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K3: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1141_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_mult_less_cancel1
thf(fact_1142_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_mult_le_cancel1
thf(fact_1143_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_1144_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_1145_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_1146_mult__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc
thf(fact_1147_not__zle__0__negative,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).

% not_zle_0_negative
thf(fact_1148_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W2: int,Z3: int] :
        ? [N3: nat] :
          ( Z3
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_1149_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K3 )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K3 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1150_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_1151_Suc__pred_H,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( N
        = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_1152_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_1153_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ ( ord_less_eq_int @ A @ B )
      | ~ ( ord_less_eq_int @ B @ A ) ) ).

% verit_la_disequality
thf(fact_1154_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_1155_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1156_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1157_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1158_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).

% one_less_mult
thf(fact_1159_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).

% n_less_m_mult_n
thf(fact_1160_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_1161_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% add_eq_if
thf(fact_1162_nat__int__comparison_I1_J,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [A3: nat,B2: nat] :
          ( ( semiri1314217659103216013at_int @ A3 )
          = ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(1)
thf(fact_1163_int__if,axiom,
    ! [P: $o,A: nat,B: nat] :
      ( ( P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
          = ( semiri1314217659103216013at_int @ A ) ) )
      & ( ~ P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
          = ( semiri1314217659103216013at_int @ B ) ) ) ) ).

% int_if
thf(fact_1164_verit__comp__simplify1_I1_J,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1165_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1166_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1167_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_1168_negative__zless__0,axiom,
    ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_1169_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N2: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% negD
thf(fact_1170_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_1171_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_1172_verit__comp__simplify1_I3_J,axiom,
    ! [B7: int,A5: int] :
      ( ( ~ ( ord_less_eq_int @ B7 @ A5 ) )
      = ( ord_less_int @ A5 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1173_verit__comp__simplify1_I3_J,axiom,
    ! [B7: nat,A5: nat] :
      ( ( ~ ( ord_less_eq_nat @ B7 @ A5 ) )
      = ( ord_less_nat @ A5 @ B7 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1174_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_1175_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_1176_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_1177_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_1178_int_Oadd_Onat__pow__Suc2,axiom,
    ! [X: int,N: nat] :
      ( ( member_int @ X @ top_top_set_int )
     => ( ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( suc @ N ) @ X )
        = ( plus_plus_int @ X @ ( add_po6256809715947028093it_nat @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ N @ X ) ) ) ) ).

% int.add.nat_pow_Suc2
thf(fact_1179_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_1180_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(5)
thf(fact_1181_int__plus,axiom,
    ! [N: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_1182_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_1183_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(7)
thf(fact_1184_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_1185_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N ) )
        = ( M = N ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_1186_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel1
thf(fact_1187_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_1188_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_1189_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_1190_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_1191_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_1192_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_1193_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel1
thf(fact_1194_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).

% int_ops(6)
thf(fact_1195_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_1196_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_1197_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_1198_zdiff__int__split,axiom,
    ! [P: int > $o,X: nat,Y: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X @ Y )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_1199_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_1200_inj__on__mult,axiom,
    ! [A: int,A2: set_int] :
      ( ( A != zero_zero_int )
     => ( inj_on_int_int @ ( times_times_int @ A ) @ A2 ) ) ).

% inj_on_mult
thf(fact_1201_inj__on__mult,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( A != zero_zero_nat )
     => ( inj_on_nat_nat @ ( times_times_nat @ A ) @ A2 ) ) ).

% inj_on_mult
thf(fact_1202_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ~ ( ord_less_nat @ T @ X6 ) ) ).

% minf(7)
thf(fact_1203_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ~ ( ord_less_int @ T @ X6 ) ) ).

% minf(7)
thf(fact_1204_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ( ord_less_nat @ X6 @ T ) ) ).

% minf(5)
thf(fact_1205_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ( ord_less_int @ X6 @ T ) ) ).

% minf(5)
thf(fact_1206_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_1207_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(4)
thf(fact_1208_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X6: nat] :
      ( ( ord_less_nat @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_1209_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X6: int] :
      ( ( ord_less_int @ X6 @ Z4 )
     => ( X6 != T ) ) ).

% minf(3)
thf(fact_1210_minf_I2_J,axiom,
    ! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z5: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z5 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z4: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z4 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P2 @ X6 )
                | ( Q3 @ X6 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1211_minf_I2_J,axiom,
    ! [P: int > $o,P2: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z5: int] :
        ! [X4: int] :
          ( ( ord_less_int @ X4 @ Z5 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z5: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z5 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z4: int] :
          ! [X6: int] :
            ( ( ord_less_int @ X6 @ Z4 )
           => ( ( ( P @ X6 )
                | ( Q @ X6 ) )
              = ( ( P2 @ X6 )
                | ( Q3 @ X6 ) ) ) ) ) ) ).

% minf(2)

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( a_r_co6205493800230438172t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ ( semiri1314217659103216013at_int @ n ) @ bot_bot_set_int ) ) @ ( semiri1314217659103216013at_int @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ x @ 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 ) ) ) @ ( genide1613390280493775889t_unit @ ( partia4118392927963588428t_unit @ top_top_set_int @ ( monoid6080699973261426200t_unit @ times_times_int @ one_one_int @ ( ring_e5272872978682396362t_unit @ zero_zero_int @ plus_plus_int @ product_Unity ) ) ) @ ( insert_int @ ( semiri1314217659103216013at_int @ n ) @ bot_bot_set_int ) ) @ ( semiri1314217659103216013at_int @ ( mult_n6028127365542633569t_unit @ ( mod_ring @ n ) @ x @ y ) ) ) ) ).

%------------------------------------------------------------------------------